How to decide when the discriminant is 0. Always be in the mood

Quadratic equations often appear during solution various tasks physics and mathematics. In this article we will look at how to solve these equalities in a universal way"through the discriminant". Examples of using the acquired knowledge are also given in the article.

What equations will we be talking about?

The figure below shows a formula in which x is an unknown variable and latin characters a, b, c represent some known numbers.

Each of these symbols is called a coefficient. As you can see, the number "a" appears before the variable x squared. This maximum degree of the presented expression, which is why it is called a quadratic equation. Its other name is often used: second-order equation. The value of a itself is square coefficient(standing with the variable squared), b is a linear coefficient (it is located next to the variable raised to the first power), finally, the number c is free member.

Note that the type of equation shown in the figure above is a general classical quadratic expression. In addition to it, there are other second-order equations in which the coefficients b and c can be zero.

When the task is set to solve the equality in question, this means that such values ​​of the variable x need to be found that would satisfy it. Here, the first thing you need to remember is the following thing: since the maximum degree of X is 2, then this type expressions cannot have more than 2 solutions. This means that if, when solving an equation, 2 values ​​of x were found that satisfy it, then you can be sure that there is no 3rd number, substituting it for x, the equality would also be true. The solutions to an equation in mathematics are called its roots.

Methods for solving second order equations

Solving equations of this type requires knowledge of some theory about them. IN school course algebras consider 4 various methods solutions. Let's list them:

• using factorization;
• using the formula for a perfect square;
• by applying the graph of the corresponding quadratic function;
• using the discriminant equation.

The advantage of the first method is its simplicity; however, it cannot be used for all equations. The second method is universal, but somewhat cumbersome. The third method is distinguished by its clarity, but it is not always convenient and applicable. And finally, using the discriminant equation is a universal and fairly simple way to find the roots of absolutely any second-order equation. Therefore, in this article we will consider only it.

Formula for obtaining the roots of the equation

Let's turn to general appearance quadratic equation. Let's write it down: a*x²+ b*x + c =0. Before using the method of solving it “through a discriminant,” you should always bring the equality to its written form. That is, it must consist of three terms (or less if b or c is 0).

For example, if there is an expression: x²-9*x+8 = -5*x+7*x², then you should first move all its terms to one side of the equality and add the terms containing the variable x in the same powers.

IN in this case this operation will lead to the following expression: -6*x²-4*x+8=0, which is equivalent to the equation 6*x²+4*x-8=0 (here we multiplied the left and right sides of the equality by -1).

In the example above, a = 6, b=4, c=-8. Note that all terms of the equality under consideration are always summed together, so if the “-” sign appears, this means that the corresponding coefficient is negative, like the number c in this case.

Having examined this point, let us now move on to the formula itself, which makes it possible to obtain the roots of a quadratic equation. It looks like the one shown in the photo below.

As can be seen from this expression, it allows you to get two roots (pay attention to the “±” sign). To do this, it is enough to substitute the coefficients b, c, and a into it.

The concept of a discriminant

In the previous paragraph, a formula was given that allows you to quickly solve any second-order equation. In it, the radical expression is called a discriminant, that is, D = b²-4*a*c.

Why is this part of the formula highlighted, and it even has proper name? The fact is that the discriminant connects all three coefficients of the equation into a single expression. Last fact means that it completely carries information about the roots, which can be expressed in the following list:

1. D>0: equality has 2 various solutions, both of which are real numbers.
2. D=0: The equation has only one root, and it is a real number.

Discriminant determination task

Let's give a simple example of how to find a discriminant. Let the following equality be given: 2*x² - 4+5*x-9*x² = 3*x-5*x²+7.

Let's bring it to standard view, we get: (2*x²-9*x²+5*x²) + (5*x-3*x) + (- 4-7) = 0, from which we come to the equality: -2*x²+2*x- 11 = 0. Here a=-2, b=2, c=-11.

Now you can use the above formula for the discriminant: D = 2² - 4*(-2)*(-11) = -84. The resulting number is the answer to the task. Since in the example the discriminant less than zero, then we can say that this quadratic equation has no real roots. Its solution will be only numbers of complex type.

An example of inequality through a discriminant

Let's solve problems of a slightly different type: given the equality -3*x²-6*x+c = 0. It is necessary to find values ​​of c for which D>0.

In this case, only 2 out of 3 coefficients are known, so it is not possible to calculate the exact value of the discriminant, but it is known that it is positive. We use the last fact when composing the inequality: D= (-6)²-4*(-3)*c>0 => 36+12*c>0. Solving the resulting inequality leads to the result: c>-3.

Let's check the resulting number. To do this, we calculate D for 2 cases: c=-2 and c=-4. The number -2 satisfies the obtained result (-2>-3), the corresponding discriminant will have the value: D = 12>0. In turn, the number -4 does not satisfy the inequality (-4. Thus, any numbers c that are greater than -3 will satisfy the condition.

An example of solving an equation

Let us present a problem that involves not only finding the discriminant, but also solving the equation. It is necessary to find the roots for the equality -2*x²+7-9*x = 0.

In this example, the discriminant is next value: D = 81-4*(-2)*7= 137. Then the roots of the equation will be determined as follows: x = (9±√137)/(-4). This exact values roots, if you calculate the root approximately, then you get the numbers: x = -5.176 and x = 0.676.

Geometric problem

We will solve a problem that will require not only the ability to calculate the discriminant, but also the application of skills abstract thinking and knowledge of how to write quadratic equations.

Bob had a 5 x 4 meter duvet. The boy wanted to sew a continuous strip of beautiful fabric to it around the entire perimeter. How thick will this strip be if we know that Bob has 10 m² of fabric.

Let the strip have a thickness of x m, then the area of ​​the fabric along the long side of the blanket will be (5+2*x)*x, and since there are 2 long sides, we have: 2*x*(5+2*x). On the short side, the area of ​​​​the sewn fabric will be 4*x, since there are 2 of these sides, we get the value 8*x. Note that 2*x was added to the long side because the length of the blanket increased by that number. The total area of ​​fabric sewn to the blanket is 10 m². Therefore, we get the equality: 2*x*(5+2*x) + 8*x = 10 => 4*x²+18*x-10 = 0.

For this example, the discriminant is equal to: D = 18²-4*4*(-10) = 484. Its root is 22. Using the formula, we find the required roots: x = (-18±22)/(2*4) = (- 5; 0.5). Obviously, of the two roots, only the number 0.5 is suitable according to the conditions of the problem.

Thus, the strip of fabric that Bob sews to his blanket will be 50 cm wide.

For example, for the trinomial \(3x^2+2x-7\), the discriminant will be equal to \(2^2-4\cdot3\cdot(-7)=4+84=88\). And for the trinomial \(x^2-5x+11\), it will be equal to \((-5)^2-4\cdot1\cdot11=25-44=-19\).

The discriminant is denoted by the letter \(D\) and is often used in solving. Also, by the value of the discriminant, you can understand what the graph approximately looks like (see below).

Discriminant and roots of the equation

The discriminant value shows the number of quadratic equations:
- if \(D\) is positive, the equation will have two roots;
- if \(D\) is equal to zero – there is only one root;
- if \(D\) is negative, there are no roots.

This does not need to be taught, it is not difficult to come to such a conclusion, simply knowing that from the discriminant (that is, \(\sqrt(D)\) is included in the formula for calculating the roots of the equation: \(x_(1)=\)\(\ frac(-b+\sqrt(D))(2a)\) and \(x_(2)=\)\(\frac(-b-\sqrt(D))(2a)\) Let's look at each case in more detail. .

If the discriminant is positive

In this case, the root of it is some positive number, which means \(x_(1)\) and \(x_(2)\) will have different meanings, because in the first formula \(\sqrt(D)\) is added, and in the second it is subtracted. And we have two different roots.

Example : Find the roots of the equation \(x^2+2x-3=0\)
Solution :

Answer : \(x_(1)=1\); \(x_(2)=-3\)

If the discriminant is zero

And how many roots will there be if the discriminant equal to zero? Let's reason.

The root formulas look like this: \(x_(1)=\)\(\frac(-b+\sqrt(D))(2a)\) and \(x_(2)=\)\(\frac(-b- \sqrt(D))(2a)\) . And if the discriminant is zero, then its root is also zero. Then it turns out:

\(x_(1)=\)\(\frac(-b+\sqrt(D))(2a)\) \(=\)\(\frac(-b+\sqrt(0))(2a)\) \(=\)\(\frac(-b+0)(2a)\) \(=\)\(\frac(-b)(2a)\)

\(x_(2)=\)\(\frac(-b-\sqrt(D))(2a)\) \(=\)\(\frac(-b-\sqrt(0))(2a) \) \(=\)\(\frac(-b-0)(2a)\) \(=\)\(\frac(-b)(2a)\)

That is, the values ​​of the roots of the equation will coincide, because adding or subtracting zero does not change anything.

Example : Find the roots of the equation \(x^2-4x+4=0\)
Solution :

Answer : \(x=2\)

Quadratic equations are studied in 8th grade, so there is nothing complicated here. The ability to solve them is absolutely necessary.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a, b and c are arbitrary numbers, and a ≠ 0.

Before studying specific solution methods, note that all quadratic equations can be divided into three classes:

1. Have no roots;
2. Have exactly one root;
3. They have two different roots.

This is important difference quadratic equations from linear ones, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.

Discriminant

Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac.

You need to know this formula by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant you can determine how many roots a quadratic equation has. Namely:

1. If D< 0, корней нет;
2. If D = 0, there is exactly one root;
3. If D > 0, there will be two roots.

Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people believe. Take a look at the examples and you will understand everything yourself:

Task. How many roots do quadratic equations have:

1. x 2 − 8x + 12 = 0;
2. 5x 2 + 3x + 7 = 0;
3. x 2 − 6x + 9 = 0.

Let's write out the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16

So the discriminant is positive, so the equation has two different roots. We analyze the second equation in a similar way:
a = 5; b = 3; c = 7;
D = 3 2 − 4 5 7 = 9 − 140 = −131.

The discriminant is negative, there are no roots. The last equation left is:
a = 1; b = −6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.

The discriminant is zero - the root will be one.

Please note that coefficients have been written down for each equation. Yes, it’s long, yes, it’s tedious, but you won’t mix up the odds and make stupid mistakes. Choose for yourself: speed or quality.

By the way, if you get the hang of it, after a while you won’t need to write down all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not that much.

Roots of a quadratic equation

Now let's move on to the solution itself. If the discriminant D > 0, the roots can be found using the formulas:

Basic formula for the roots of a quadratic equation

When D = 0, you can use any of these formulas - you will get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.

1. x 2 − 2x − 3 = 0;
2. 15 − 2x − x 2 = 0;
3. x 2 + 12x + 36 = 0.

First equation:
x 2 − 2x − 3 = 0 ⇒ a = 1; b = −2; c = −3;
D = (−2) 2 − 4 1 (−3) = 16.

D > 0 ⇒ the equation has two roots. Let's find them:

Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 · (−1) · 15 = 64.

D > 0 ⇒ the equation again has two roots. Let's find them

\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]

Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.

D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:

As you can see from the examples, everything is very simple. If you know the formulas and can count, there will be no problems. Most often, errors occur when substituting negative coefficients into the formula. Here again, the technique described above will help: look at the formula literally, write down each step - and very soon you will get rid of errors.

Incomplete quadratic equations

It happens that a quadratic equation is slightly different from what is given in the definition. For example:

1. x 2 + 9x = 0;
2. x 2 − 16 = 0.

It is easy to notice that these equations are missing one of the terms. Such quadratic equations are even easier to solve than standard ones: they don’t even require calculating the discriminant. So, let's introduce a new concept:

The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.

Of course, a very difficult case is possible when both of these coefficients are equal to zero: b = c = 0. In this case, the equation takes the form ax 2 = 0. Obviously, such an equation has a single root: x = 0.

Let's consider the remaining cases. Let b = 0, then we get an incomplete quadratic equation of the form ax 2 + c = 0. Let’s transform it a little:

Since arithmetic square root exists only from non-negative number, the last equality makes sense only for (−c /a) ≥ 0. Conclusion:

1. If in an incomplete quadratic equation of the form ax 2 + c = 0 the inequality (−c /a) ≥ 0 is satisfied, there will be two roots. The formula is given above;
2. If (−c /a)< 0, корней нет.

As you can see, the discriminant was not required - in incomplete quadratic equations there is no complex calculations. In fact, it is not even necessary to remember the inequality (−c /a) ≥ 0. It is enough to express the value x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If it is negative, there will be no roots at all.

Now let's look at equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factor the polynomial:

The product is zero when at least one of the factors is zero. This is where the roots come from. In conclusion, let’s look at a few of these equations:

Task. Solve quadratic equations:

1. x 2 − 7x = 0;
2. 5x 2 + 30 = 0;
3. 4x 2 − 9 = 0.

x 2 − 7x = 0 ⇒ x · (x − 7) = 0 ⇒ x 1 = 0; x 2 = −(−7)/1 = 7.

5x 2 + 30 = 0 ⇒ 5x 2 = −30 ⇒ x 2 = −6. There are no roots, because a square cannot be equal to a negative number.

4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 = −1.5.

Discriminant is a multi-valued term. In this article we will talk about the discriminant of a polynomial, which allows you to determine whether a given polynomial has valid solutions. The formula for the quadratic polynomial is found in the school course on algebra and analysis. How to find a discriminant? What is needed to solve the equation?

A quadratic polynomial or equation of the second degree is called i * w ^ 2 + j * w + k equals 0, where "i" and "j" are the first and second coefficients, respectively, "k" is a constant, sometimes called the "dismissive term", and "w" is a variable. Its roots will be all the values ​​of the variable at which it turns into an identity. Such an equality can be rewritten as the product of i, (w - w1) and (w - w2) equal to 0. In this case, it is obvious that if the coefficient “i” does not become zero, then the function on the left side will become zero only if if x takes the value w1 or w2. These values ​​are the result of setting the polynomial equal to zero.

To find the value of a variable at which quadratic polynomial becomes zero, an auxiliary construction is used, built on its coefficients and called the discriminant. This design is calculated according to the formula D equals j * j - 4 * i * k. Why is it used?

1. She says are there any valid results.
2. She helps calculate them.

How does this value show the presence of real roots:

• If it is positive, then we can find two roots in the region real numbers.
• If the discriminant is zero, then both solutions are the same. We can say that there is only one solution, and it is from the field of real numbers.
• If the discriminant is less than zero, then the polynomial has no real roots.

Calculation options for securing material

For the sum (7 * w^2; 3 * w; 1) equal to 0 We calculate D using the formula 3 * 3 - 4 * 7 * 1 = 9 - 28, we get -19. A discriminant value below zero indicates that there are no results on the actual line.

If we consider 2 * w^2 - 3 * w + 1 equivalent to 0, then D is calculated as (-3) squared minus the product of numbers (4; 2; 1) and equals 9 - 8, that is, 1. Positive value says there are two results on the real line.

If we take the sum (w ^ 2; 2 * w; 1) and equate it to 0, D is calculated as two squared minus the product of the numbers (4; 1; 1). This expression will simplify to 4 - 4 and go to zero. It turns out that the results are the same. If you look closely at this formula, then it will become clear that this is “ perfect square" This means that the equality can be rewritten in the form (w + 1) ^ 2 = 0. It became obvious that the result in this problem is “-1”. In a situation where D is 0, left side Equalities can always be collapsed using the “square of the sum” formula.

Using a discriminant in calculating roots

This auxiliary construction not only shows the number of real solutions, but also helps to find them. General formula The calculation for the second degree equation is:

w = (-j +/- d) / (2 * i), where d is the discriminant to the power of 1/2.

Let's say the discriminant is below zero, then d is imaginary and the results are imaginary.

D is zero, then d equal to D to the power of 1/2 is also zero. Solution: -j / (2 * i). Again considering 1 * w ^ 2 + 2 * w + 1 = 0, we find results equivalent to -2 / (2 * 1) = -1.

Suppose D > 0, then d - real number, and the answer here breaks down into two parts: w1 = (-j + d) / (2 * i) and w2 = (-j - d) / (2 * i). Both results will be valid. Let's look at 2 * w ^ 2 - 3 * w + 1 = 0. Here the discriminant and d are ones. It turns out that w1 is equal to (3 + 1) divided by (2 * 2) or 1, and w2 is equal to (3 - 1) divided by 2 * 2 or 1/2.

Equation result quadratic expression to zero is calculated according to the algorithm:

1. Determination of quantity valid solutions.
2. Calculation d = D^(1/2).
3. Finding the result according to the formula (-j +/- d) / (2 * i).
4. Substituting the obtained result into the original equality for verification.

Some special cases

Depending on the coefficients, the solution may be somewhat simplified. Obviously, if the coefficient of a variable to the second power is zero, then a linear equality is obtained. When the coefficient of a variable to the first power is zero, then two options are possible:

1. the polynomial is expanded into a difference of squares when the free term is negative;
2. for a positive constant, no real solutions can be found.

If the free term is zero, then the roots will be (0; -j)

But there are other special cases that simplify finding a solution.

Reduced second degree equation

The given is called such quadratic trinomial, where the coefficient in front of the leading term is one. For this situation, Vieta’s theorem is applicable, which states that the sum of the roots is equal to the coefficient of the variable to the first power, multiplied by -1, and the product corresponds to the constant “k”.

Therefore, w1 + w2 equals -j and w1 * w2 equals k if the first coefficient is one. To verify the correctness of this representation, you can express w2 = -j - w1 from the first formula and substitute it into the second equality w1 * (-j - w1) = k. The result is the original equality w1 ^ 2 + j * w1 + k = 0.

Important to note, that i * w ^ 2 + j * w + k = 0 can be achieved by dividing by “i”. The result will be: w^2 + j1 * w + k1 = 0, where j1 is equal to j/i and k1 is equal to k/i.

Let's look at the already solved 2 * w^2 - 3 * w + 1 = 0 with the results w1 = 1 and w2 = 1/2. We need to divide it in half, as a result w ^ 2 - 3/2 * w + 1/2 = 0. Let's check that the conditions of the theorem are true for the results found: 1 + 1/2 = 3/2 and 1*1/2 = 1 /2.

Even second factor

If the factor of a variable to the first power (j) is divisible by 2, then it will be possible to simplify the formula and look for a solution through a quarter of the discriminant D/4 = (j / 2) ^ 2 - i * k. it turns out w = (-j +/- d/2) / i, where d/2 = D/4 to the power of 1/2.

If i = 1, and the coefficient j is even, then the solution will be the product of -1 and half the coefficient of the variable w, plus/minus the root of the square of this half minus the constant “k”. Formula: w = -j/2 +/- (j^2/4 - k)^1/2.

Higher discriminant order

The discriminant of the second degree trinomial discussed above is the most commonly used special case. In the general case, the discriminant of a polynomial is multiplied squares of the differences of the roots of this polynomial. Therefore, a discriminant equal to zero indicates the presence of at least two multiple solutions.

Consider i * w^3 + j * w^2 + k * w + m = 0.

D = j^2 * k^2 - 4 * i * k^3 - 4 * i^3 * k - 27 * i^2 * m^2 + 18 * i * j * k * m.

Suppose the discriminant exceeds zero. This means that there are three roots in the region of real numbers. At zero there are multiple solutions. If D< 0, то два корня комплексно-сопряженные, которые дают negative value when squaring, and also one root is real.

Video

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