This topic may seem complicated at first due to many not so simple formulas. Not only do the quadratic equations themselves have long notations, but the roots are also found through the discriminant. In total, three new formulas are obtained. Not very easy to remember. This is possible only after solving such equations frequently. Then all the formulas will be remembered by themselves.

General view of a quadratic equation

Here we propose their explicit recording, when the largest degree is written first, and then in descending order. There are often situations when the terms are inconsistent. Then it is better to rewrite the equation in descending order of the degree of the variable.

Let us introduce some notation. They are presented in the table below.

If we accept these notations, all quadratic equations are reduced to the following notation.

Moreover, the coefficient a ≠ 0. Let this formula be designated number one.

When an equation is given, it is not clear how many roots there will be in the answer. Because one of three options is always possible:

• the solution will have two roots;
• the answer will be one number;
• the equation will have no roots at all.

And until the decision is finalized, it is difficult to understand which option will appear in a particular case.

Types of recordings of quadratic equations

There may be different entries in tasks. They will not always look like the general quadratic equation formula. Sometimes it will be missing some terms. What was written above is the complete equation. If you remove the second or third term in it, you get something else. These records are also called quadratic equations, only incomplete.

Moreover, only terms with coefficients “b” and “c” can disappear. The number "a" cannot be equal to zero under any circumstances. Because in this case the formula turns into a linear equation. The formulas for the incomplete form of equations will be as follows:

So, there are only two types; in addition to complete ones, there are also incomplete quadratic equations. Let the first formula be number two, and the second - three.

Discriminant and dependence of the number of roots on its value

You need to know this number in order to calculate the roots of the equation. It can always be calculated, no matter what the formula of the quadratic equation is. In order to calculate the discriminant, you need to use the equality written below, which will have number four.

After substituting the coefficient values ​​into this formula, you can get numbers with different signs. If the answer is yes, then the answer to the equation will be two different roots. If the number is negative, there will be no roots of the quadratic equation. If it is equal to zero, there will be only one answer.

How to solve a complete quadratic equation?

In fact, consideration of this issue has already begun. Because first you need to find a discriminant. After it is determined that there are roots of the quadratic equation, and their number is known, you need to use formulas for the variables. If there are two roots, then you need to apply the following formula.

Since it contains a “±” sign, there will be two meanings. The expression under the square root sign is the discriminant. Therefore, the formula can be rewritten differently.

Formula number five. From the same record it is clear that if the discriminant is equal to zero, then both roots will take the same values.

If solving quadratic equations has not yet been worked out, then it is better to write down the values ​​of all coefficients before applying the discriminant and variable formulas. Later this moment will not cause difficulties. But at the very beginning there is confusion.

How to solve an incomplete quadratic equation?

Everything is much simpler here. There is not even a need for additional formulas. And those that have already been written down for the discriminant and the unknown will not be needed.

First, let's look at incomplete equation number two. In this equality, it is necessary to take the unknown quantity out of brackets and solve the linear equation, which will remain in brackets. The answer will have two roots. The first one is necessarily equal to zero, because there is a multiplier consisting of the variable itself. The second one will be obtained by solving a linear equation.

Incomplete equation number three is solved by moving the number from the left side of the equality to the right. Then you need to divide by the coefficient facing the unknown. All that remains is to extract the square root and remember to write it down twice with opposite signs.

Below are some steps that will help you learn how to solve all kinds of equalities that turn into quadratic equations. They will help the student to avoid mistakes due to inattention. These shortcomings can cause poor grades when studying the extensive topic “Quadratic Equations (Grade 8).” Subsequently, these actions will not need to be performed constantly. Because a stable skill will appear.

• First you need to write the equation in standard form. That is, first the term with the largest degree of the variable, and then - without a degree, and last - just a number.

• If a minus appears before the coefficient “a”, it can complicate the work for a beginner studying quadratic equations. It's better to get rid of it. For this purpose, all equality must be multiplied by “-1”. This means that all terms will change sign to the opposite.

• It is recommended to get rid of fractions in the same way. Simply multiply the equation by the appropriate factor so that the denominators cancel out.

Examples

It is required to solve the following quadratic equations:

x 2 − 7x = 0;

15 − 2x − x 2 = 0;

x 2 + 8 + 3x = 0;

12x + x 2 + 36 = 0;

(x+1) 2 + x + 1 = (x+1)(x+2).

The first equation: x 2 − 7x = 0. It is incomplete, therefore it is solved as described for formula number two.

After taking it out of brackets, it turns out: x (x - 7) = 0.

The first root takes the value: x 1 = 0. The second will be found from the linear equation: x - 7 = 0. It is easy to see that x 2 = 7.

Second equation: 5x 2 + 30 = 0. Again incomplete. Only it is solved as described for the third formula.

After moving 30 to the right side of the equation: 5x 2 = 30. Now you need to divide by 5. It turns out: x 2 = 6. The answers will be the numbers: x 1 = √6, x 2 = - √6.

The third equation: 15 − 2x − x 2 = 0. Hereinafter, solving quadratic equations will begin by rewriting them in standard form: − x 2 − 2x + 15 = 0. Now it’s time to use the second useful tip and multiply everything by minus one . It turns out x 2 + 2x - 15 = 0. Using the fourth formula, you need to calculate the discriminant: D = 2 2 - 4 * (- 15) = 4 + 60 = 64. It is a positive number. From what is said above, it turns out that the equation has two roots. They need to be calculated using the fifth formula. It turns out that x = (-2 ± √64) / 2 = (-2 ± 8) / 2. Then x 1 = 3, x 2 = - 5.

The fourth equation x 2 + 8 + 3x = 0 is transformed into this: x 2 + 3x + 8 = 0. Its discriminant is equal to this value: -23. Since this number is negative, the answer to this task will be the following entry: “There are no roots.”

The fifth equation 12x + x 2 + 36 = 0 should be rewritten as follows: x 2 + 12x + 36 = 0. After applying the formula for the discriminant, the number zero is obtained. This means that it will have one root, namely: x = -12/ (2 * 1) = -6.

The sixth equation (x+1) 2 + x + 1 = (x+1)(x+2) requires transformations, which consist in the fact that you need to bring similar terms, first opening the brackets. In place of the first there will be the following expression: x 2 + 2x + 1. After the equality, this entry will appear: x 2 + 3x + 2. After similar terms are counted, the equation will take the form: x 2 - x = 0. It has become incomplete . Something similar to this has already been discussed a little higher. The roots of this will be the numbers 0 and 1.

Yakupova M.I. 1

Smirnova Yu.V. 1

1 Municipal budgetary educational institution secondary school No. 11

The text of the work is posted without images and formulas.
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History of quadratic equations

Babylon

The need to solve equations not only of the first degree, but also of the second, in ancient times was caused by the need to solve problems related to finding the areas of land plots, with the development of astronomy and mathematics itself. Quadratic equations could be solved around 2000 BC. e. Babylonians. The rules for solving these equations set out in the Babylonian texts are essentially the same as modern ones, but these texts lack the concept of a negative number and general methods for solving quadratic equations.

Ancient Greece

In Ancient Greece, scientists such as Diophantus, Euclid and Heron also worked on solving quadratic equations. Diophantus Diophantus of Alexandria is an ancient Greek mathematician who presumably lived in the 3rd century AD. The main work of Diophantus is “Arithmetic” in 13 books. Euclid. Euclid is an ancient Greek mathematician, the author of the first theoretical treatise on mathematics that has come down to us, Heron. Heron - Greek mathematician and engineer first in Greece in the 1st century AD. gives a purely algebraic way to solve a quadratic equation

India

Problems on quadratic equations are found already in the astronomical treatise “Aryabhattiam”, compiled in 499 by the Indian mathematician and astronomer Aryabhatta. Another Indian scientist, Brahmagupta (VII century), outlined the general rule for solving quadratic equations reduced to a single canonical form: ax2 + bx = c, a> 0. (1) In equation (1) the coefficients can be negative. Brahmagupta's rule is essentially the same as ours. Public competitions in solving difficult problems were common in India. One of the old Indian books says the following about such competitions: “As the sun outshines the stars with its brilliance, so a learned man will outshine his glory in public assemblies by proposing and solving algebraic problems.” Problems were often presented in poetic form.

This is one of the problems of the famous Indian mathematician of the 12th century. Bhaskars.

“A flock of frisky monkeys

And twelve along the vines, having eaten to my heart’s content, had fun

They began to jump, hanging

Part eight of them squared

How many monkeys were there?

I was having fun in the clearing

Tell me, in this pack?

Bhaskara's solution indicates that the author knew that the roots of quadratic equations are two-valued. Bhaskar writes the equation corresponding to the problem as x2 - 64x = - 768 and, in order to complete the left side of this equation to a square, adds 322 to both sides, then obtaining: x2 - b4x + 322 = -768 + 1024, (x - 32)2 = 256, x - 32= ±16, x1 = 16, x2 = 48.

Quadratic equations in 17th century Europe

Formulas for solving quadratic equations along the lines of Al-Khorezmi in Europe were first set forth in the Book of Abacus, written in 1202 by the Italian mathematician Leonardo Fibonacci. This voluminous work, which reflects the influence of mathematics, both from the countries of Islam and from ancient Greece, is distinguished by its completeness and clarity of presentation. The author independently developed some new algebraic examples of solving problems and was the first in Europe to approach the introduction of negative numbers. His book contributed to the spread of algebraic knowledge not only in Italy, but also in Germany, France and other European countries. Many problems from the Book of Abacus were used in almost all European textbooks of the 16th - 17th centuries. and partly XVIII. The derivation of the formula for solving a quadratic equation in general form is available from Vieth, but Vieth recognized only positive roots. Italian mathematicians Tartaglia, Cardano, Bombelli were among the first in the 16th century. In addition to positive ones, negative roots are also taken into account. Only in the 17th century. Thanks to the work of Girard, Descartes, Newton and other scientists, the method of solving quadratic equations takes on a modern form.

Definition of a quadratic equation

An equation of the form ax 2 + bx + c = 0, where a, b, c are numbers, is called quadratic.

Quadratic equation coefficients

Numbers a, b, c are the coefficients of the quadratic equation. a is the first coefficient (before x²), a ≠ 0; b is the second coefficient (before x); c is the free term (without x).

Which of these equations are not quadratic??

1. 4x² + 4x + 1 = 0;2. 5x - 7 = 0;3. - x² - 5x - 1 = 0;4. 2/x² + 3x + 4 = 0;5. ¼ x² - 6x + 1 = 0;6. 2x² = 0;

7. 4x² + 1 = 0;8. x² - 1/x = 0;9. 2x² - x = 0;10. x² -16 = 0;11. 7x² + 5x = 0;12. -8x²= 0;13. 5x³ +6x -8= 0.

Types of quadratic equations

Reduced is a quadratic equation in which the leading coefficient is equal to one. Such an equation can be obtained by dividing the entire expression by the leading coefficient a:

x 2 + px + q =0, p = b/a, q = c/a

A quadratic equation is called complete if all its coefficients are nonzero.

A quadratic equation is called incomplete in which at least one of the coefficients, except the leading one (either the second coefficient or the free term), is equal to zero.

Methods for solving quadratic equations

Method I General formula for calculating roots

To find the roots of a quadratic equation ax 2 + b + c = 0 In general, you should use the algorithm below:

Calculate the value of the discriminant of a quadratic equation: this is the expression for it D= b 2 - 4ac

Derivation of the formula:

Note: It is obvious that the formula for a root of multiplicity 2 is a special case of the general formula, obtained by substituting the equality D=0 into it, and the conclusion about the absence of real roots at D0, and (displaystyle (sqrt (-1))=i) = i.

The presented method is universal, but it is far from the only one. Solving a single equation can be approached in a variety of ways, with preferences usually depending on the solver. In addition, often for this purpose some of the methods turn out to be much more elegant, simple, and less labor-intensive than the standard one.

II method. Roots of a quadratic equation with an even coefficient b III method. Solving incomplete quadratic equations

IV method. Using partial ratios of coefficients

There are special cases of quadratic equations in which the coefficients are in relationships with each other, making them much easier to solve.

Roots of a quadratic equation in which the sum of the leading coefficient and the free term is equal to the second coefficient

If in a quadratic equation ax 2 + bx + c = 0 the sum of the first coefficient and the free term is equal to the second coefficient: a+b=c, then its roots are -1 and the number opposite to the ratio of the free term to the leading coefficient ( -c/a).

Hence, before solving any quadratic equation, you should check the possibility of applying this theorem to it: compare the sum of the leading coefficient and the free term with the second coefficient.

Roots of a quadratic equation whose sum of all coefficients is zero

If in a quadratic equation the sum of all its coefficients is zero, then the roots of such an equation are 1 and the ratio of the free term to the leading coefficient ( c/a).

Hence, before solving an equation using standard methods, you should check the applicability of this theorem to it: add up all the coefficients of this equation and see if this sum is not equal to zero.

V method. Factoring a quadratic trinomial into linear factors

If the trinomial is of the form (displaystyle ax^(2)+bx+c(anot =0))ax 2 + bx + c(a ≠ 0) can somehow be represented as a product of linear factors (displaystyle (kx+m)(lx+n)=0)(kx + m)(lx + n), then we can find the roots of the equation ax 2 + bx + c = 0- they will be -m/k and n/l, indeed, after all (displaystyle (kx+m)(lx+n)=0Longleftrightarrow kx+m=0cup lx+n=0)(kx + m)(lx + n) = 0 kx + mUlx + n, and having solved the indicated linear equations, we obtain the above. Note that the quadratic trinomial does not always decompose into linear factors with real coefficients: this is possible if the corresponding equation has real roots.

Let's consider some special cases

Using the squared sum (difference) formula

If the quadratic trinomial has the form (displaystyle (ax)^(2)+2abx+b^(2))ax 2 + 2abx + b 2 , then by applying the above formula to it, we can factor it into linear factors and, therefore, find roots:

(ax) 2 + 2abx + b 2 = (ax + b) 2

Isolating the full square of the sum (difference)

The above formula is also used using a method called “selecting the full square of the sum (difference).” In relation to the above quadratic equation with the previously introduced notation, this means the following:

Note: If you notice, this formula coincides with the one proposed in the section “Roots of the reduced quadratic equation”, which, in turn, can be obtained from the general formula (1) by substituting the equality a=1. This fact is not just a coincidence: using the described method, albeit with some additional reasoning, one can derive a general formula and also prove the properties of the discriminant.

VI method. Using the direct and inverse Vieta theorem

Vieta's direct theorem (see below in the section of the same name) and its inverse theorem allow you to solve the above quadratic equations orally, without resorting to rather cumbersome calculations using formula (1).

According to the converse theorem, every pair of numbers (number) (displaystyle x_(1),x_(2))x 1, x 2, being a solution to the system of equations below, are the roots of the equation

In the general case, that is, for an unreduced quadratic equation ax 2 + bx + c = 0

x 1 + x 2 = -b/a, x 1 * x 2 = c/a

A direct theorem will help you find numbers that satisfy these equations orally. With its help, you can determine the signs of the roots without knowing the roots themselves. To do this, you should follow the rule:

1) if the free term is negative, then the roots have different signs, and the largest in absolute value of the roots has a sign opposite to the sign of the second coefficient of the equation;

2) if the free term is positive, then both roots have the same sign, and this is the sign opposite to the sign of the second coefficient.

VII method. Transfer method

The so-called “transfer” method allows you to reduce the solution of unreduced and irreducible equations to the form of reduced equations with integer coefficients by dividing them by the leading coefficient to the solution of reduced equations with integer coefficients. It is as follows:

Next, the equation is solved orally in the manner described above, then they return to the original variable and find the roots of the equations (displaystyle y_(1)=ax_(1)) y 1 =ax 1 And y 2 =ax 2 .(displaystyle y_(2)=ax_(2))

Geometric meaning

The graph of a quadratic function is a parabola. The solutions (roots) of a quadratic equation are the abscissas of the points of intersection of the parabola with the abscissa axis. If the parabola described by a quadratic function does not intersect the x-axis, the equation has no real roots. If a parabola intersects the x-axis at one point (at the vertex of the parabola), the equation has one real root (the equation is also said to have two coinciding roots). If the parabola intersects the x-axis at two points, the equation has two real roots (see image on the right.)

If coefficient (displaystyle a) a positive, the branches of the parabola are directed upward and vice versa. If the coefficient (displaystyle b) bpositive (if positive (displaystyle a) a, if negative, vice versa), then the vertex of the parabola lies in the left half-plane and vice versa.

Application of quadratic equations in life

The quadratic equation is widely used. It is used in many calculations, structures, sports, and also around us.

Let us consider and give some examples of the application of the quadratic equation.

Sport. High jumps: during the jumper's run-up, calculations related to the parabola are used to get the most accurate shot at the take-off bar and fly high.

Also, similar calculations are needed in throwing. The flight range of an object depends on the quadratic equation.

Astronomy. The trajectory of the planets can be found using a quadratic equation.

Airplane flight. Airplane takeoff is the main component of flight. Here we take the calculation for low resistance and acceleration of takeoff.

Quadratic equations are also used in various economic disciplines, in programs for processing audio, video, vector and raster graphics.

Conclusion

As a result of the work done, it turned out that quadratic equations attracted scientists back in ancient times; they had already encountered them when solving some problems and tried to solve them. Looking at different ways to solve quadratic equations, I came to the conclusion that not all of them are simple. In my opinion, the best way to solve quadratic equations is to solve them using formulas. The formulas are easy to remember, this method is universal. The hypothesis that equations are widely used in life and mathematics was confirmed. After studying the topic, I learned many interesting facts about quadratic equations, their use, application, types, solutions. And I will be happy to continue studying them. I hope this will help me do well in my exams.

List of used literature

Site materials:

Wikipedia

Open lesson.rf

Handbook of Elementary Mathematics Vygodsky M. Ya.

Quadratic equation problems are studied both in the school curriculum and in universities. They mean equations of the form a*x^2 + b*x + c = 0, where x- variable, a, b, c – constants; a<>0 . The task is to find the roots of the equation.

Geometric meaning of quadratic equation

The graph of a function that is represented by a quadratic equation is a parabola. The solutions (roots) of a quadratic equation are the points of intersection of the parabola with the abscissa (x) axis. It follows that there are three possible cases:
1) the parabola has no points of intersection with the abscissa axis. This means that it is in the upper plane with branches up or the bottom with branches down. In such cases, the quadratic equation has no real roots (it has two complex roots).

2) the parabola has one point of intersection with the Ox axis. Such a point is called the vertex of the parabola, and the quadratic equation at it acquires its minimum or maximum value. In this case, the quadratic equation has one real root (or two identical roots).

3) The last case is more interesting in practice - there are two points of intersection of the parabola with the abscissa axis. This means that there are two real roots of the equation.

Based on the analysis of the coefficients of the powers of the variables, interesting conclusions can be drawn about the placement of the parabola.

1) If the coefficient a is greater than zero, then the parabola’s branches are directed upward; if it is negative, the parabola’s branches are directed downward.

2) If the coefficient b is greater than zero, then the vertex of the parabola lies in the left half-plane, if it takes a negative value, then in the right half-plane.

Derivation of the formula for solving a quadratic equation

Let's transfer the constant from the quadratic equation

for the equal sign, we get the expression

Multiply both sides by 4a

To get a complete square on the left, add b^2 on both sides and carry out the transformation

From here we find

Formula for the discriminant and roots of a quadratic equation

The discriminant is the value of the radical expression. If it is positive, then the equation has two real roots, calculated by the formula When the discriminant is zero, the quadratic equation has one solution (two coinciding roots), which can be easily obtained from the above formula for D=0. When the discriminant is negative, the equation has no real roots. However, solutions to the quadratic equation are found in the complex plane, and their value is calculated using the formula

Vieta's theorem

Let's consider two roots of a quadratic equation and construct a quadratic equation on their basis. Vieta's theorem itself easily follows from the notation: if we have a quadratic equation of the form then the sum of its roots is equal to the coefficient p taken with the opposite sign, and the product of the roots of the equation is equal to the free term q. The formula for the above will look like If in a classical equation the constant a is nonzero, then you need to divide the entire equation by it, and then apply Vieta’s theorem.

Factoring quadratic equation schedule

Let the task be set: factor a quadratic equation. To do this, we first solve the equation (find the roots). Next, we substitute the found roots into the expansion formula for the quadratic equation. This will solve the problem.

Quadratic equation problems

Task 1. Find the roots of a quadratic equation

x^2-26x+120=0 .

Solution: Write down the coefficients and substitute them into the discriminant formula

The root of this value is 14, it is easy to find with a calculator, or remember with frequent use, however, for convenience, at the end of the article I will give you a list of squares of numbers that can often be encountered in such problems.
We substitute the found value into the root formula

and we get

Task 2. Solve the equation

2x 2 +x-3=0.

Solution: We have a complete quadratic equation, write out the coefficients and find the discriminant


Using known formulas we find the roots of the quadratic equation

Task 3. Solve the equation

9x 2 -12x+4=0.

Solution: We have a complete quadratic equation. Determining the discriminant

We got a case where the roots coincide. Find the values ​​of the roots using the formula

Task 4. Solve the equation

x^2+x-6=0 .

Solution: In cases where there are small coefficients for x, it is advisable to apply Vieta’s theorem. By its condition we obtain two equations

From the second condition we find that the product must be equal to -6. This means that one of the roots is negative. We have the following possible pair of solutions (-3;2), (3;-2) . Taking into account the first condition, we reject the second pair of solutions.
The roots of the equation are equal

Problem 5. Find the lengths of the sides of a rectangle if its perimeter is 18 cm and its area is 77 cm 2.

Solution: Half the perimeter of a rectangle is equal to the sum of its adjacent sides. Let's denote x as the larger side, then 18-x is its smaller side. The area of ​​the rectangle is equal to the product of these lengths:
x(18-x)=77;
or
x 2 -18x+77=0.
Let's find the discriminant of the equation

Calculating the roots of the equation

If x=11, That 18's=7 , the opposite is also true (if x=7, then 21's=9).

Problem 6. Factor the quadratic equation 10x 2 -11x+3=0.

Solution: Let's calculate the roots of the equation, to do this we find the discriminant

We substitute the found value into the root formula and calculate

We apply the formula for decomposing a quadratic equation by roots

Opening the brackets we obtain an identity.

Quadratic equation with parameter

Example 1. At what parameter values A , does the equation (a-3)x 2 + (3-a)x-1/4=0 have one root?

Solution: By direct substitution of the value a=3 we see that it has no solution. Next, we will use the fact that with a zero discriminant the equation has one root of multiplicity 2. Let's write out the discriminant

Let's simplify it and equate it to zero

We have obtained a quadratic equation with respect to the parameter a, the solution of which can be easily obtained using Vieta’s theorem. The sum of the roots is 7, and their product is 12. By simple search we establish that the numbers 3,4 will be the roots of the equation. Since we already rejected the solution a=3 at the beginning of the calculations, the only correct one will be - a=4. Thus, when a=4 the equation has one root.

Example 2. At what parameter values A , equation a(a+3)x^2+(2a+6)x-3a-9=0 has more than one root?

Solution: Let's first consider the singular points, they will be the values ​​a=0 and a=-3. When a=0, the equation will be simplified to the form 6x-9=0; x=3/2 and there will be one root. For a= -3 we obtain the identity 0=0.
Let's calculate the discriminant

and find the value of a at which it is positive

From the first condition we get a>3. For the second, we find the discriminant and roots of the equation


Let us determine the intervals where the function takes positive values. By substituting the point a=0 we get 3>0 . So, outside the interval (-3;1/3) the function is negative. Don't forget the point a=0, which should be excluded because the original equation has one root in it.
As a result, we obtain two intervals that satisfy the conditions of the problem

There will be many similar tasks in practice, try to figure out the tasks yourself and do not forget to take into account the conditions that are mutually exclusive. Study well the formulas for solving quadratic equations; they are often needed in calculations in various problems and sciences.

We continue to study the topic “ solving equations" We have already become acquainted with linear equations and are moving on to getting acquainted with quadratic equations.

First, we will look at what a quadratic equation is, how it is written in general form, and give related definitions. After this, we will use examples to examine in detail how incomplete quadratic equations are solved. Next, we will move on to solving complete equations, obtain the root formula, get acquainted with the discriminant of a quadratic equation, and consider solutions to typical examples. Finally, let's trace the connections between the roots and coefficients.

Page navigation.

What is a quadratic equation? Their types

First you need to clearly understand what a quadratic equation is. Therefore, it is logical to start a conversation about quadratic equations with the definition of a quadratic equation, as well as related definitions. After this, you can consider the main types of quadratic equations: reduced and unreduced, as well as complete and incomplete equations.

Definition and examples of quadratic equations

Definition.

Quadratic equation is an equation of the form a x 2 +b x+c=0, where x is a variable, a, b and c are some numbers, and a is non-zero.

Let's say right away that quadratic equations are often called equations of the second degree. This is due to the fact that the quadratic equation is algebraic equation second degree.

The stated definition allows us to give examples of quadratic equations. So 2 x 2 +6 x+1=0, 0.2 x 2 +2.5 x+0.03=0, etc. These are quadratic equations.

Definition.

Numbers a, b and c are called coefficients of the quadratic equation a·x 2 +b·x+c=0, and coefficient a is called the first, or the highest, or the coefficient of x 2, b is the second coefficient, or the coefficient of x, and c is the free term.

For example, let's take a quadratic equation of the form 5 x 2 −2 x −3=0, here the leading coefficient is 5, the second coefficient is equal to −2, and the free term is equal to −3. Please note that when the coefficients b and/or c are negative, as in the example just given, the short form of the quadratic equation is 5 x 2 −2 x−3=0 , rather than 5 x 2 +(−2 )·x+(−3)=0 .

It is worth noting that when the coefficients a and/or b are equal to 1 or −1, then they are usually not explicitly present in the quadratic equation, which is due to the peculiarities of writing such. For example, in the quadratic equation y 2 −y+3=0 the leading coefficient is one, and the coefficient of y is equal to −1.

Reduced and unreduced quadratic equations

Depending on the value of the leading coefficient, reduced and unreduced quadratic equations are distinguished. Let us give the corresponding definitions.

Definition.

A quadratic equation in which the leading coefficient is 1 is called given quadratic equation. Otherwise the quadratic equation is untouched.

According to this definition, quadratic equations x 2 −3·x+1=0, x 2 −x−2/3=0, etc. – given, in each of them the first coefficient is equal to one. A 5 x 2 −x−1=0, etc. - unreduced quadratic equations, their leading coefficients are different from 1.

From any unreduced quadratic equation, by dividing both sides by the leading coefficient, you can go to the reduced one. This action is an equivalent transformation, that is, the reduced quadratic equation obtained in this way has the same roots as the original unreduced quadratic equation, or, like it, has no roots.

Let us look at an example of how the transition from an unreduced quadratic equation to a reduced one is performed.

Example.

From the equation 3 x 2 +12 x−7=0, go to the corresponding reduced quadratic equation.

Solution.

We just need to divide both sides of the original equation by the leading coefficient 3, it is non-zero, so we can perform this action. We have (3 x 2 +12 x−7):3=0:3, which is the same, (3 x 2):3+(12 x):3−7:3=0, and then (3:3) x 2 +(12:3) x−7:3=0, from where . This is how we obtained the reduced quadratic equation, which is equivalent to the original one.

Answer:

Complete and incomplete quadratic equations

The definition of a quadratic equation contains the condition a≠0. This condition is necessary so that the equation a x 2 + b x + c = 0 is quadratic, since when a = 0 it actually becomes a linear equation of the form b x + c = 0.

As for the coefficients b and c, they can be equal to zero, both individually and together. In these cases, the quadratic equation is called incomplete.

Definition.

The quadratic equation a x 2 +b x+c=0 is called incomplete, if at least one of the coefficients b, c is equal to zero.

In turn

Definition.

Complete quadratic equation is an equation in which all coefficients are different from zero.

Such names were not given by chance. This will become clear from the following discussions.

If the coefficient b is zero, then the quadratic equation takes the form a·x 2 +0·x+c=0, and it is equivalent to the equation a·x 2 +c=0. If c=0, that is, the quadratic equation has the form a·x 2 +b·x+0=0, then it can be rewritten as a·x 2 +b·x=0. And with b=0 and c=0 we get the quadratic equation a·x 2 =0. The resulting equations differ from the complete quadratic equation in that their left-hand sides do not contain either a term with the variable x, or a free term, or both. Hence their name - incomplete quadratic equations.

So the equations x 2 +x+1=0 and −2 x 2 −5 x+0.2=0 are examples of complete quadratic equations, and x 2 =0, −2 x 2 =0, 5 x 2 +3=0 , −x 2 −5 x=0 are incomplete quadratic equations.

Solving incomplete quadratic equations

From the information in the previous paragraph it follows that there is three types of incomplete quadratic equations:

• a·x 2 =0, the coefficients b=0 and c=0 correspond to it;
• a x 2 +c=0 when b=0 ;
• and a·x 2 +b·x=0 when c=0.

Let us examine in order how incomplete quadratic equations of each of these types are solved.

a x 2 =0

Let's start with solving incomplete quadratic equations in which the coefficients b and c are equal to zero, that is, with equations of the form a x 2 =0. The equation a·x 2 =0 is equivalent to the equation x 2 =0, which is obtained from the original by dividing both parts by a non-zero number a. Obviously, the root of the equation x 2 =0 is zero, since 0 2 =0. This equation has no other roots, which is explained by the fact that for any non-zero number p the inequality p 2 >0 holds, which means that for p≠0 the equality p 2 =0 is never achieved.

So, the incomplete quadratic equation a·x 2 =0 has a single root x=0.

As an example, we give the solution to the incomplete quadratic equation −4 x 2 =0. It is equivalent to the equation x 2 =0, its only root is x=0, therefore, the original equation has a single root zero.

A short solution in this case can be written as follows:
−4 x 2 =0 ,
x 2 =0,
x=0 .

a x 2 +c=0

Now let's look at how incomplete quadratic equations are solved in which the coefficient b is zero and c≠0, that is, equations of the form a x 2 +c=0. We know that moving a term from one side of the equation to the other with the opposite sign, as well as dividing both sides of the equation by a non-zero number, gives an equivalent equation. Therefore, we can carry out the following equivalent transformations of the incomplete quadratic equation a x 2 +c=0:

• move c to the right side, which gives the equation a x 2 =−c,
• and divide both sides by a, we get .

The resulting equation allows us to draw conclusions about its roots. Depending on the values ​​of a and c, the value of the expression can be negative (for example, if a=1 and c=2, then ) or positive (for example, if a=−2 and c=6, then ), it is not zero , since by condition c≠0. We will separately analyze the cases and.

If , then the equation has no roots. This statement follows from the fact that the square of any number is a non-negative number. It follows from this that when , then for any number p the equality cannot be true.

If , then the situation with the roots of the equation is different. In this case, if we remember about , then the root of the equation immediately becomes obvious; it is the number, since . It’s easy to guess that the number is also the root of the equation, indeed, . This equation has no other roots, which can be shown, for example, by contradiction. Let's do this.

Let us denote the roots of the equation just announced as x 1 and −x 1 . Suppose that the equation has one more root x 2, different from the indicated roots x 1 and −x 1. It is known that substituting its roots into an equation instead of x turns the equation into a correct numerical equality. For x 1 and −x 1 we have , and for x 2 we have . The properties of numerical equalities allow us to perform term-by-term subtraction of correct numerical equalities, so subtracting the corresponding parts of the equalities gives x 1 2 −x 2 2 =0. The properties of operations with numbers allow us to rewrite the resulting equality as (x 1 −x 2)·(x 1 +x 2)=0. We know that the product of two numbers is equal to zero if and only if at least one of them is equal to zero. Therefore, from the resulting equality it follows that x 1 −x 2 =0 and/or x 1 +x 2 =0, which is the same, x 2 =x 1 and/or x 2 =−x 1. So we came to a contradiction, since at the beginning we said that the root of the equation x 2 is different from x 1 and −x 1. This proves that the equation has no roots other than and .

Let us summarize the information in this paragraph. The incomplete quadratic equation a x 2 +c=0 is equivalent to the equation that

• has no roots if ,
• has two roots and , if .

Let's consider examples of solving incomplete quadratic equations of the form a·x 2 +c=0.

Let's start with the quadratic equation 9 x 2 +7=0. After moving the free term to the right side of the equation, it will take the form 9 x 2 =−7. Dividing both sides of the resulting equation by 9, we arrive at . Since the right side has a negative number, this equation has no roots, therefore, the original incomplete quadratic equation 9 x 2 +7 = 0 has no roots.

Let's solve another incomplete quadratic equation −x 2 +9=0. We move the nine to the right side: −x 2 =−9. Now we divide both sides by −1, we get x 2 =9. On the right side there is a positive number, from which we conclude that or . Then we write down the final answer: the incomplete quadratic equation −x 2 +9=0 has two roots x=3 or x=−3.

a x 2 +b x=0

It remains to deal with the solution of the last type of incomplete quadratic equations for c=0. Incomplete quadratic equations of the form a x 2 + b x = 0 allows you to solve factorization method. Obviously, we can, located on the left side of the equation, for which it is enough to take the common factor x out of brackets. This allows us to move from the original incomplete quadratic equation to an equivalent equation of the form x·(a·x+b)=0. And this equation is equivalent to a set of two equations x=0 and a·x+b=0, the latter of which is linear and has a root x=−b/a.

So, the incomplete quadratic equation a·x 2 +b·x=0 has two roots x=0 and x=−b/a.

To consolidate the material, we will analyze the solution to a specific example.

Example.

Solve the equation.

Solution.

Taking x out of brackets gives the equation . It is equivalent to two equations x=0 and . We solve the resulting linear equation: , and by dividing the mixed number by an ordinary fraction, we find . Therefore, the roots of the original equation are x=0 and .

After gaining the necessary practice, solutions to such equations can be written briefly:

Answer:

x=0 , .

Discriminant, formula for the roots of a quadratic equation

To solve quadratic equations, there is a root formula. Let's write it down formula for the roots of a quadratic equation: , Where D=b 2 −4 a c- so-called discriminant of a quadratic equation. The entry essentially means that .

It is useful to know how the root formula was derived and how it is used in finding the roots of quadratic equations. Let's figure this out.

Derivation of the formula for the roots of a quadratic equation

Let us need to solve the quadratic equation a·x 2 +b·x+c=0. Let's perform some equivalent transformations:

• We can divide both sides of this equation by a non-zero number a, resulting in the following quadratic equation.
• Now select a complete square on its left side: . After this, the equation will take the form .
• At this stage, it is possible to transfer the last two terms to the right side with the opposite sign, we have .
• And let’s also transform the expression on the right side: .

As a result, we arrive at an equation that is equivalent to the original quadratic equation a·x 2 +b·x+c=0.

We have already solved equations similar in form in the previous paragraphs, when we examined. This allows us to draw the following conclusions regarding the roots of the equation:

• if , then the equation has no real solutions;
• if , then the equation has the form , therefore, , from which its only root is visible;
• if , then or , which is the same as or , that is, the equation has two roots.

Thus, the presence or absence of roots of the equation, and therefore the original quadratic equation, depends on the sign of the expression on the right side. In turn, the sign of this expression is determined by the sign of the numerator, since the denominator 4·a 2 is always positive, that is, by the sign of the expression b 2 −4·a·c. This expression b 2 −4 a c was called discriminant of a quadratic equation and designated by the letter D. From here the essence of the discriminant is clear - based on its value and sign, they conclude whether the quadratic equation has real roots, and if so, what is their number - one or two.

Let's return to the equation and rewrite it using the discriminant notation: . And we draw conclusions:

• if D<0 , то это уравнение не имеет действительных корней;
• if D=0, then this equation has a single root;
• finally, if D>0, then the equation has two roots or, which can be rewritten in the form or, and after expanding and bringing the fractions to a common denominator we obtain.

So we derived the formulas for the roots of the quadratic equation, they have the form , where the discriminant D is calculated by the formula D=b 2 −4·a·c.

With their help, with a positive discriminant, you can calculate both real roots of a quadratic equation. When the discriminant is zero, both formulas give the same value of the root, corresponding to a unique solution to the quadratic equation. And with a negative discriminant, when we try to use the formula for the roots of a quadratic equation, we are faced with extracting the square root of a negative number, which takes us beyond the scope of the school curriculum. With a negative discriminant, the quadratic equation has no real roots, but has a pair complex conjugate roots, which can be found using the same root formulas we obtained.

Algorithm for solving quadratic equations using root formulas

In practice, when solving quadratic equations, you can immediately use the root formula to calculate their values. But this is more related to finding complex roots.

However, in a school algebra course we usually talk not about complex, but about real roots of a quadratic equation. In this case, it is advisable, before using the formulas for the roots of a quadratic equation, to first find the discriminant, make sure that it is non-negative (otherwise, we can conclude that the equation does not have real roots), and only then calculate the values ​​of the roots.

The above reasoning allows us to write algorithm for solving a quadratic equation. To solve the quadratic equation a x 2 +b x+c=0, you need to:

• using the discriminant formula D=b 2 −4·a·c, calculate its value;
• conclude that a quadratic equation has no real roots if the discriminant is negative;
• calculate the only root of the equation using the formula if D=0;
• find two real roots of a quadratic equation using the root formula if the discriminant is positive.

Here we just note that if the discriminant is equal to zero, you can also use the formula; it will give the same value as .

You can move on to examples of using the algorithm for solving quadratic equations.

Examples of solving quadratic equations

Let's consider solutions to three quadratic equations with a positive, negative and zero discriminant. Having dealt with their solution, by analogy it will be possible to solve any other quadratic equation. Let's begin.

Example.

Find the roots of the equation x 2 +2·x−6=0.

Solution.

In this case, we have the following coefficients of the quadratic equation: a=1, b=2 and c=−6. According to the algorithm, you first need to calculate the discriminant; to do this, we substitute the indicated a, b and c into the discriminant formula, we have D=b 2 −4·a·c=2 2 −4·1·(−6)=4+24=28. Since 28>0, that is, the discriminant is greater than zero, the quadratic equation has two real roots. Let's find them using the root formula, we get , here you can simplify the resulting expressions by doing moving the multiplier beyond the root sign followed by reduction of the fraction:

Answer:

Let's move on to the next typical example.

Example.

Solve the quadratic equation −4 x 2 +28 x−49=0 .

Solution.

We start by finding the discriminant: D=28 2 −4·(−4)·(−49)=784−784=0. Therefore, this quadratic equation has a single root, which we find as , that is,

Answer:

x=3.5.

It remains to consider solving quadratic equations with a negative discriminant.

Example.

Solve the equation 5·y 2 +6·y+2=0.

Solution.

Here are the coefficients of the quadratic equation: a=5, b=6 and c=2. We substitute these values ​​into the discriminant formula, we have D=b 2 −4·a·c=6 2 −4·5·2=36−40=−4. The discriminant is negative, therefore, this quadratic equation has no real roots.

If you need to indicate complex roots, then we apply the well-known formula for the roots of a quadratic equation, and perform operations with complex numbers:

Answer:

there are no real roots, complex roots are: .

Let us note once again that if the discriminant of a quadratic equation is negative, then in school they usually immediately write down an answer in which they indicate that there are no real roots, and complex roots are not found.

Root formula for even second coefficients

The formula for the roots of a quadratic equation, where D=b 2 −4·a·c allows you to obtain a formula of a more compact form, allowing you to solve quadratic equations with an even coefficient for x (or simply with a coefficient of the form 2·n, for example, or 14· ln5=2·7·ln5 ). Let's get her out.

Let's say we need to solve a quadratic equation of the form a x 2 +2 n x+c=0. Let's find its roots using the formula we know. To do this, we calculate the discriminant D=(2 n) 2 −4 a c=4 n 2 −4 a c=4 (n 2 −a c), and then we use the root formula:

Let us denote the expression n 2 −a c as D 1 (sometimes it is denoted D "). Then the formula for the roots of the quadratic equation under consideration with the second coefficient 2 n will take the form , where D 1 =n 2 −a·c.

It is easy to see that D=4·D 1, or D 1 =D/4. In other words, D 1 is the fourth part of the discriminant. It is clear that the sign of D 1 is the same as the sign of D . That is, the sign D 1 is also an indicator of the presence or absence of roots of a quadratic equation.

So, to solve a quadratic equation with a second coefficient 2·n, you need

• Calculate D 1 =n 2 −a·c ;
• If D 1<0 , то сделать вывод, что действительных корней нет;
• If D 1 =0, then calculate the only root of the equation using the formula;
• If D 1 >0, then find two real roots using the formula.

Let's consider solving the example using the root formula obtained in this paragraph.

Example.

Solve the quadratic equation 5 x 2 −6 x −32=0 .

Solution.

The second coefficient of this equation can be represented as 2·(−3) . That is, you can rewrite the original quadratic equation in the form 5 x 2 +2 (−3) x−32=0, here a=5, n=−3 and c=−32, and calculate the fourth part of the discriminant: D 1 =n 2 −a·c=(−3) 2 −5·(−32)=9+160=169. Since its value is positive, the equation has two real roots. Let's find them using the appropriate root formula:

Note that it was possible to use the usual formula for the roots of a quadratic equation, but in this case more computational work would have to be performed.

Answer:

Simplifying the form of quadratic equations

Sometimes, before starting to calculate the roots of a quadratic equation using formulas, it doesn’t hurt to ask the question: “Is it possible to simplify the form of this equation?” Agree that in terms of calculations it will be easier to solve the quadratic equation 11 x 2 −4 x−6=0 than 1100 x 2 −400 x−600=0.

Typically, simplifying the form of a quadratic equation is achieved by multiplying or dividing both sides by a certain number. For example, in the previous paragraph it was possible to simplify the equation 1100 x 2 −400 x −600=0 by dividing both sides by 100.

A similar transformation is carried out with quadratic equations, the coefficients of which are not . In this case, both sides of the equation are usually divided by the absolute values ​​of its coefficients. For example, let's take the quadratic equation 12 x 2 −42 x+48=0. absolute values ​​of its coefficients: GCD(12, 42, 48)= GCD(GCD(12, 42), 48)= GCD(6, 48)=6. Dividing both sides of the original quadratic equation by 6, we arrive at the equivalent quadratic equation 2 x 2 −7 x+8=0.

And multiplying both sides of a quadratic equation is usually done to get rid of fractional coefficients. In this case, multiplication is carried out by the denominators of its coefficients. For example, if both sides of the quadratic equation are multiplied by LCM(6, 3, 1)=6, then it will take the simpler form x 2 +4·x−18=0.

In conclusion of this point, we note that they almost always get rid of the minus at the highest coefficient of a quadratic equation by changing the signs of all terms, which corresponds to multiplying (or dividing) both sides by −1. For example, usually one moves from the quadratic equation −2 x 2 −3 x+7=0 to the solution 2 x 2 +3 x−7=0 .

Relationship between roots and coefficients of a quadratic equation

The formula for the roots of a quadratic equation expresses the roots of the equation through its coefficients. Based on the root formula, you can obtain other relationships between roots and coefficients.

The most well-known and applicable formulas from Vieta’s theorem are of the form and . In particular, for the given quadratic equation, the sum of the roots is equal to the second coefficient with the opposite sign, and the product of the roots is equal to the free term. For example, by the form of the quadratic equation 3 x 2 −7 x + 22 = 0 we can immediately say that the sum of its roots is equal to 7/3, and the product of the roots is equal to 22/3.

Using the already written formulas, you can obtain a number of other connections between the roots and coefficients of the quadratic equation. For example, you can express the sum of the squares of the roots of a quadratic equation through its coefficients: .

References.

• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.

Entry level

Quadratic equations. The Comprehensive Guide (2019)

In the term “quadratic equation,” the key word is “quadratic.” This means that the equation must necessarily contain a variable (that same x) squared, and there should not be xes to the third (or greater) power.

The solution of many equations comes down to solving quadratic equations.

Let's learn to determine that this is a quadratic equation and not some other equation.

Example 1.

Let's get rid of the denominator and multiply each term of the equation by

Let's move everything to the left side and arrange the terms in descending order of powers of X

Now we can say with confidence that this equation is quadratic!

Example 2.

Multiply the left and right sides by:

This equation, although it was originally in it, is not quadratic!

Example 3.

Let's multiply everything by:

Scary? The fourth and second degrees... However, if we make a replacement, we will see that we have a simple quadratic equation:

Example 4.

It seems to be there, but let's take a closer look. Let's move everything to the left side:

See, it's reduced - and now it's a simple linear equation!

Now try to determine for yourself which of the following equations are quadratic and which are not:

Examples:

Answers:

1. square;
2. square;
3. not square;
4. not square;
5. not square;
6. square;
7. not square;
8. square.

Mathematicians conventionally divide all quadratic equations into the following types:

• Complete quadratic equations- equations in which the coefficients and, as well as the free term c, are not equal to zero (as in the example). In addition, among complete quadratic equations there are given- these are equations in which the coefficient (the equation from example one is not only complete, but also reduced!)

• Incomplete quadratic equations- equations in which the coefficient and or the free term c are equal to zero:

  They are incomplete because they are missing some element. But the equation must always contain X squared!!! Otherwise, it will no longer be a quadratic equation, but some other equation.

Why did they come up with such a division? It would seem that there is an X squared, and okay. This division is determined by the solution methods. Let's look at each of them in more detail.

Solving incomplete quadratic equations

First, let's focus on solving incomplete quadratic equations - they are much simpler!

There are types of incomplete quadratic equations:

1. , in this equation the coefficient is equal.
2. , in this equation the free term is equal to.
3. , in this equation the coefficient and the free term are equal.

1. i. Since we know how to take the square root, let's use this equation to express

The expression can be either negative or positive. A squared number cannot be negative, because when multiplying two negative or two positive numbers, the result will always be a positive number, so: if, then the equation has no solutions.

And if, then we get two roots. These formulas do not need to be memorized. The main thing is that you must know and always remember that it cannot be less.

Let's try to solve some examples.

Example 5:

Solve the equation

Now all that remains is to extract the root from the left and right sides. After all, you remember how to extract roots?

Answer:

Never forget about roots with a negative sign!!!

Example 6:

Solve the equation

Answer:

Example 7:

Solve the equation

Oh! The square of a number cannot be negative, which means that the equation

no roots!

For such equations that have no roots, mathematicians came up with a special icon - (empty set). And the answer can be written like this:

Answer:

Thus, this quadratic equation has two roots. There are no restrictions here, since we did not extract the root.
Example 8:

Solve the equation

Let's take the common factor out of brackets:

Thus,

This equation has two roots.

Answer:

The simplest type of incomplete quadratic equations (although they are all simple, right?). Obviously, this equation always has only one root:

We will dispense with examples here.

Solving complete quadratic equations

We remind you that a complete quadratic equation is an equation of the form equation where

Solving complete quadratic equations is a little more difficult (just a little) than these.

Remember Any quadratic equation can be solved using a discriminant! Even incomplete.

The other methods will help you do it faster, but if you have problems with quadratic equations, first master the solution using the discriminant.

1. Solving quadratic equations using a discriminant.

Solving quadratic equations using this method is very simple; the main thing is to remember the sequence of actions and a couple of formulas.

If, then the equation has a root. You need to pay special attention to the step. Discriminant () tells us the number of roots of the equation.

• If, then the formula in the step will be reduced to. Thus, the equation will only have a root.
• If, then we will not be able to extract the root of the discriminant at the step. This indicates that the equation has no roots.

Let's go back to our equations and look at some examples.

Example 9:

Solve the equation

Step 1 we skip.

Step 2.

We find the discriminant:

This means the equation has two roots.

Step 3.

Answer:

Example 10:

Solve the equation

The equation is presented in standard form, so Step 1 we skip.

Step 2.

We find the discriminant:

This means that the equation has one root.

Answer:

Example 11:

Solve the equation

The equation is presented in standard form, so Step 1 we skip.

Step 2.

We find the discriminant:

This means we will not be able to extract the root of the discriminant. There are no roots of the equation.

Now we know how to correctly write down such answers.

Answer: no roots

2. Solving quadratic equations using Vieta's theorem.

If you remember, there is a type of equation that is called reduced (when the coefficient a is equal to):

Such equations are very easy to solve using Vieta’s theorem:

Sum of roots given quadratic equation is equal, and the product of the roots is equal.

Example 12:

Solve the equation

This equation can be solved using Vieta's theorem because .

The sum of the roots of the equation is equal, i.e. we get the first equation:

And the product is equal to:

Let's compose and solve the system:

• And. The amount is equal to;
• And. The amount is equal to;
• And. The amount is equal.

and are the solution to the system:

Answer: ; .

Example 13:

Solve the equation

Answer:

Example 14:

Solve the equation

The equation is given, which means:

Answer:

QUADRATE EQUATIONS. MIDDLE LEVEL

What is a quadratic equation?

In other words, a quadratic equation is an equation of the form, where - the unknown, - some numbers, and.

The number is called the highest or first coefficient quadratic equation, - second coefficient, A - free member.

Why? Because if the equation immediately becomes linear, because will disappear.

In this case, and can be equal to zero. In this chair equation is called incomplete. If all the terms are in place, that is, the equation is complete.

Solutions to various types of quadratic equations

Methods for solving incomplete quadratic equations:

First, let's look at methods for solving incomplete quadratic equations - they are simpler.

We can distinguish the following types of equations:

I., in this equation the coefficient and the free term are equal.

II. , in this equation the coefficient is equal.

III. , in this equation the free term is equal to.

Now let's look at the solution to each of these subtypes.

Obviously, this equation always has only one root:

A squared number cannot be negative, because when you multiply two negative or two positive numbers, the result will always be a positive number. That's why:

if, then the equation has no solutions;

if we have two roots

These formulas do not need to be memorized. The main thing to remember is that it cannot be less.

Examples:

Solutions:

Answer:

Never forget about roots with a negative sign!

The square of a number cannot be negative, which means that the equation

no roots.

To briefly write down that a problem has no solutions, we use the empty set icon.

Answer:

So, this equation has two roots: and.

Answer:

Let's take the common factor out of brackets:

The product is equal to zero if at least one of the factors is equal to zero. This means that the equation has a solution when:

So, this quadratic equation has two roots: and.

Example:

Solve the equation.

Solution:

Let's factor the left side of the equation and find the roots:

Answer:

Methods for solving complete quadratic equations:

1. Discriminant

Solving quadratic equations this way is easy, the main thing is to remember the sequence of actions and a couple of formulas. Remember, any quadratic equation can be solved using a discriminant! Even incomplete.

Did you notice the discriminant root in the formula for roots? But the discriminant can be negative. What to do? We need to pay special attention to step 2. The discriminant tells us the number of roots of the equation.

• If, then the equation has roots:
• If, then the equation has the same roots, and in fact, one root:

  Such roots are called double roots.

• If, then the root of the discriminant is not extracted. This indicates that the equation has no roots.

Why are different numbers of roots possible? Let us turn to the geometric meaning of the quadratic equation. The graph of the function is a parabola:

In a special case, which is a quadratic equation, . This means that the roots of a quadratic equation are the points of intersection with the abscissa axis (axis). A parabola may not intersect the axis at all, or may intersect it at one (when the vertex of the parabola lies on the axis) or two points.

In addition, the coefficient is responsible for the direction of the branches of the parabola. If, then the branches of the parabola are directed upward, and if, then downward.

Examples:

Solutions:

Answer:

Answer: .

Answer:

This means there are no solutions.

Answer: .

2. Vieta's theorem

It is very easy to use Vieta's theorem: you just need to choose a pair of numbers whose product is equal to the free term of the equation, and the sum is equal to the second coefficient taken with the opposite sign.

It is important to remember that Vieta's theorem can only be applied in reduced quadratic equations ().

Let's look at a few examples:

Example #1:

Solve the equation.

Solution:

This equation can be solved using Vieta's theorem because . Other coefficients: ; .

The sum of the roots of the equation is:

And the product is equal to:

Let's select pairs of numbers whose product is equal and check whether their sum is equal:

• And. The amount is equal to;
• And. The amount is equal to;
• And. The amount is equal.

and are the solution to the system:

Thus, and are the roots of our equation.

Answer: ; .

Example #2:

Solution:

Let's select pairs of numbers that give in the product, and then check whether their sum is equal:

and: they give in total.

and: they give in total. To obtain, it is enough to simply change the signs of the supposed roots: and, after all, the product.

Answer:

Example #3:

Solution:

The free term of the equation is negative, and therefore the product of the roots is a negative number. This is only possible if one of the roots is negative and the other is positive. Therefore the sum of the roots is equal to differences of their modules.

Let us select such pairs of numbers that give in the product, and the difference of which is equal to:

and: their difference is equal - does not fit;

and: - not suitable;

and: - not suitable;

and: - suitable. All that remains is to remember that one of the roots is negative. Since their sum must be equal, the root with a smaller modulus must be negative: . We check:

Answer:

Example #4:

Solve the equation.

Solution:

The equation is given, which means:

The free term is negative, and therefore the product of the roots is negative. And this is only possible when one root of the equation is negative and the other is positive.

Let's select pairs of numbers whose product is equal, and then determine which roots should have a negative sign:

Obviously, only the roots and are suitable for the first condition:

Answer:

Example #5:

Solve the equation.

Solution:

The equation is given, which means:

The sum of the roots is negative, which means that at least one of the roots is negative. But since their product is positive, it means both roots have a minus sign.

Let us select pairs of numbers whose product is equal to:

Obviously, the roots are the numbers and.

Answer:

Agree, it’s very convenient to come up with roots orally, instead of counting this nasty discriminant. Try to use Vieta's theorem as often as possible.

But Vieta’s theorem is needed in order to facilitate and speed up finding the roots. In order for you to benefit from using it, you must bring the actions to automaticity. And for this, solve five more examples. But don't cheat: you can't use a discriminant! Only Vieta's theorem:

Solutions to tasks for independent work:

Task 1. ((x)^(2))-8x+12=0

According to Vieta's theorem:

As usual, we start the selection with the piece:

Not suitable because the amount;

: the amount is just what you need.

Answer: ; .

Task 2.

And again our favorite Vieta theorem: the sum must be equal, and the product must be equal.

But since it must be not, but, we change the signs of the roots: and (in total).

Answer: ; .

Task 3.

Hmm... Where is that?

You need to move all the terms into one part:

The sum of the roots is equal to the product.

Okay, stop! The equation is not given. But Vieta's theorem is applicable only in the given equations. So first you need to give an equation. If you can’t lead, give up this idea and solve in another way (for example, through a discriminant). Let me remind you that to give a quadratic equation means to make the leading coefficient equal:

Great. Then the sum of the roots is equal to and the product.

Here it’s as easy as shelling pears to choose: after all, it’s a prime number (sorry for the tautology).

Answer: ; .

Task 4.

The free member is negative. What's special about this? And the fact is that the roots will have different signs. And now, during the selection, we check not the sum of the roots, but the difference in their modules: this difference is equal, but a product.

So, the roots are equal to and, but one of them is minus. Vieta's theorem tells us that the sum of the roots is equal to the second coefficient with the opposite sign, that is. This means that the smaller root will have a minus: and, since.

Answer: ; .

Task 5.

What should you do first? That's right, give the equation:

Again: we select the factors of the number, and their difference should be equal to:

The roots are equal to and, but one of them is minus. Which? Their sum should be equal, which means that the minus will have a larger root.

Answer: ; .

Let me summarize:

1. Vieta's theorem is used only in the quadratic equations given.
2. Using Vieta's theorem, you can find the roots by selection, orally.
3. If the equation is not given or no suitable pair of factors of the free term is found, then there are no whole roots, and you need to solve it in another way (for example, through a discriminant).

3. Method for selecting a complete square

If all terms containing the unknown are represented in the form of terms from abbreviated multiplication formulas - the square of the sum or difference - then after replacing the variables, the equation can be presented in the form of an incomplete quadratic equation of the type.

For example:

Example 1:

Solve the equation: .

Solution:

Answer:

Example 2:

Solve the equation: .

Solution:

Answer:

In general, the transformation will look like this:

It follows: .

Doesn't remind you of anything? This is a discriminatory thing! That's exactly how we got the discriminant formula.

QUADRATE EQUATIONS. BRIEFLY ABOUT THE MAIN THINGS

Quadratic equation- this is an equation of the form, where - the unknown, - the coefficients of the quadratic equation, - the free term.

Complete quadratic equation- an equation in which the coefficients are not equal to zero.

Reduced quadratic equation- an equation in which the coefficient, that is: .

Incomplete quadratic equation- an equation in which the coefficient and or the free term c are equal to zero:

• if the coefficient, the equation looks like: ,
• if there is a free term, the equation has the form: ,
• if and, the equation looks like: .

1. Algorithm for solving incomplete quadratic equations

1.1. Incomplete quadratic equation of the form, where, :

1) Let’s express the unknown: ,

2) Check the sign of the expression:

• if, then the equation has no solutions,
• if, then the equation has two roots.

1.2. Incomplete quadratic equation of the form, where, :

1) Let’s take the common factor out of brackets: ,

2) The product is equal to zero if at least one of the factors is equal to zero. Therefore, the equation has two roots:

1.3. An incomplete quadratic equation of the form, where:

This equation always has only one root: .

2. Algorithm for solving complete quadratic equations of the form where

2.1. Solution using discriminant

1) Let's bring the equation to standard form: ,

2) Let's calculate the discriminant using the formula: , which indicates the number of roots of the equation:

3) Find the roots of the equation:

• if, then the equation has roots, which are found by the formula:
• if, then the equation has a root, which is found by the formula:
• if, then the equation has no roots.

2.2. Solution using Vieta's theorem

The sum of the roots of the reduced quadratic equation (equation of the form where) is equal, and the product of the roots is equal, i.e. , A.

2.3. Solution by the method of selecting a complete square