The topic “Square trinomial and its roots” is studied in the 9th grade algebra course. Like any other mathematics lesson, a lesson on this topic requires special teaching tools and methods. Visibility is necessary. This includes this video lesson, which was designed specifically to make the teacher’s work easier.
This lesson lasts 6:36 minutes. During this time, the author manages to reveal the topic completely. The teacher will only have to select tasks on the topic to reinforce the material.
The lesson begins by showing examples of polynomials with one variable. Then the definition of the root of the polynomial appears on the screen. This definition is supported by an example where it is necessary to find the roots of a polynomial. Having solved the equation, the author obtains the roots of the polynomial.
The following is a remark that quadratic trinomials also include those polynomials of the second degree in which the second, third, or both coefficients, except the leading one, are equal to zero. This information is supported by an example where the free coefficient is zero.
The author then explains how to find the roots of a quadratic trinomial. To do this, you need to solve a quadratic equation. And the author suggests checking this using an example where a quadratic trinomial is given. We need to find its roots. The solution is constructed based on the solution to the quadratic equation obtained from the given quadratic trinomial. The solution is written on the screen in detail, clearly and understandably. While solving this example, the author remembers how to solve a quadratic equation, writes down the formulas, and gets the result. The answer is recorded on the screen.
The author explained finding the roots of a square trinomial based on an example. When students understand the essence, they can move on to more general points, which is what the author does. Therefore, he further summarizes all of the above. In general terms in mathematical language, the author writes down the rule for finding the roots of a square trinomial.
The following is a remark that in some problems it is more convenient to write the quadratic trinomial a little differently. This entry is shown on the screen. That is, it turns out that from a square trinomial one can extract the square of a binomial. It is proposed to consider such a transformation with an example. The solution to this example is shown on the screen. As in the previous example, the solution is constructed in detail with all the necessary explanations. The author then considers a problem that uses the information just given. This is a geometric proof problem. The solution contains an illustration in the form of a drawing. The solution to the problem is described in detail and clearly.
This concludes the lesson. But the teacher can select tasks based on the students’ abilities that will correspond to the given topic.
This video lesson can be used as an explanation of new material in algebra lessons. It is perfect for students to independently prepare for the lesson.
Factoring quadratic trinomials is one of the school assignments that everyone faces sooner or later. How to do it? What is the formula for factoring a quadratic trinomial? Let's figure it out step by step using examples.
General formula
Quadratic trinomials are factorized by solving a quadratic equation. This is a simple problem that can be solved by several methods - by finding the discriminant using Vieta's theorem, there is also a graphical solution. The first two methods are studied in high school.
The general formula looks like this:lx 2 +kx+n=l(x-x 1)(x-x 2) (1)
Algorithm for completing the task
In order to factor quadratic trinomials, you need to know Vita's theorem, have a solution program at hand, be able to find a solution graphically, or look for roots of a second-degree equation using the discriminant formula. If a quadratic trinomial is given and it needs to be factorized, the algorithm is as follows:
1) Equate the original expression to zero to obtain an equation.
2) Give similar terms (if necessary).
3) Find the roots using any known method. The graphical method is best used if it is known in advance that the roots are integers and small numbers. It must be remembered that the number of roots is equal to the maximum degree of the equation, that is, the quadratic equation has two roots.
4) Substitute the value X into expression (1).
5) Write down the factorization of quadratic trinomials.
Examples
Practice allows you to finally understand how this task is performed. The following examples illustrate the factorization of a quadratic trinomial:
it is necessary to expand the expression:
Let's resort to our algorithm:
1) x 2 -17x+32=0
2) similar terms are reduced
3) using Vieta’s formula, it is difficult to find roots for this example, so it is better to use the expression for the discriminant:
D=289-128=161=(12.69) 2
4) Let’s substitute the roots we found into the basic formula for decomposition:
(x-2.155) * (x-14.845)
5) Then the answer will be like this:
x 2 -17x+32=(x-2.155)(x-14.845)
Let's check whether the solutions found by the discriminant correspond to the Vieta formulas:
14,845 . 2,155=32
For these roots, Vieta’s theorem is applied, they were found correctly, which means the factorization we obtained is also correct.
Similarly, we expand 12x 2 + 7x-6.
x 1 =-7+(337) 1/2
x 2 =-7-(337)1/2
In the previous case, the solutions were non-integer, but real numbers, which are easy to find if you have a calculator in front of you. Now let's look at a more complex example, in which the roots will be complex: factor x 2 + 4x + 9. Using Vieta's formula, the roots cannot be found, and the discriminant is negative. The roots will be on the complex plane.
D=-20
Based on this, we obtain the roots that interest us -4+2i*5 1/2 and -4-2i * 5 1/2 since (-20) 1/2 = 2i*5 1/2 .
We obtain the desired decomposition by substituting the roots into the general formula.
Another example: you need to factor the expression 23x 2 -14x+7.
We have the equation 23x 2 -14x+7 =0
D=-448
This means the roots are 14+21.166i and 14-21.166i. The answer will be:
23x 2 -14x+7 =23(x- 14-21,166i )*(X- 14+21,166i ).
Let us give an example that can be solved without the help of a discriminant.
Let's say we need to expand the quadratic equation x 2 -32x+255. Obviously, it can also be solved using a discriminant, but in this case it is faster to find the roots.
x 1 =15
x 2 =17
Means x 2 -32x+255 =(x-15)(x-17).
Presentation for a mathematics lesson in 9th grade on the topic “Square trinomial and its roots” with the content of tasks at an in-depth level of studying the subject. The presentation is designed for continuous use throughout the lesson. Assignments of various types in content.
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Item of the plan Item of the plan Item of the plan Item of the plan Item of the plan Updating knowledge Studying the topic of the lesson Encyclopedic reference Dynamic minute Homework The square trinomial and its roots were prepared by the mathematics teacher: 1KK Radchenko Natalya Fedorovna
Updating knowledge Studying the topic of the lesson Encyclopedic reference Dynamic minute Homework Updating knowledge ◊ 1 Repetition of material about functions; ◊ 2 Theoretical foundations for solving a quadratic equation; ◊ 3 Vieta’s theorem; ◊ 4 Total.
Updating knowledge Repetition of material: among these functions, indicate linear decreasing functions: y= x²+12 y= -x-24 y= 9x+8 h= 23-23x h= 1/x² g= (x+16)² g= - 3
Updating knowledge How is the presence and number of roots of a quadratic equation determined? How to calculate the discriminant of a quadratic equation D = 2. Name the formulas for the roots of the quadratic equation D>0, then x 1,2 = D = 0, then x =
Updating knowledge t² - 2t – 3 = 0 3. Calculate the discriminant and answer the question “How many roots does the quadratic equation have?” D= 16 >0, two roots What is the product of the roots? X 1 x 2 = - 3 5. What is the sum of the roots of the equation? X 1 + x 2 = 2 6. What can be said about the signs of the roots? Roots of different signs 7. Find the roots by selection. X 1 = 3, x 2 = -1
Studying the topic of the lesson ◊ 1 Reporting the topic of the lesson; ◊ 2 Theoretical foundations of the concept “Square trinomial and its roots”; ◊ 3 Statements of great thinkers about mathematics; ◊ 4 Analysis of examples of topics; Studying the topic of the lesson Encyclopedic reference Dynamic minute Homework
Square trinomial and its roots A square trinomial is a polynomial of the form ax² + bx + c, where x is a variable, a, b and c are some numbers, and a≠ 0. The root of a quadratic trinomial is the value of a variable at which the value of this trinomial is zero. To find the roots of the quadratic trinomial ax² + bx + c, you need to solve the quadratic equation ax² + bx + c =0
The square trinomial and its roots It is not enough to have a good mind, the main thing is to use it well. R. Descartes Everyone should be able to think consistently, judge with evidence, and refute incorrect conclusions: a physicist and a poet, a tractor driver and a chemist. E. Kolman
Encyclopedic reference ◊ 1 The concept of “parameter”; ◊ 2 The meaning of the word “parameter” in Russian dictionaries and a dictionary of foreign words; ◊ 3 Designation and scope of application of the parameter; ◊ 4 Examples with parameters. Encyclopedic reference Dynamic minute Homework
Encyclopedic reference PARAMETER (from the Greek παραμετρέω - I measure, leaving). A quantity included in a mathematical formula and maintaining a constant value within one phenomenon or for a given particular task..., (mat.) Parameter is a constant value, expressed by a letter, retaining its constant value only under the conditions of a given task... “Dictionary of foreign words.” 3. At what value of the parameter m does the square trinomial 2x ² + 2тх – m – 0.5 have a single root? Find this root.
Dynamic pause ◊ 1 Solving a “problem problem”; ◊ 2 Historical background: letter from the past; Dynamic Minute Homework
Dynamic pause At what value of the parameter t does the square trinomial 2х ² + 2тх – т – 0.5 = 0 and have a single root? Find this root. The quadratic equation has one root D=0 D= b² - 4ac; a=2, b=2m, c= - m – 0.5 D= (2m)² - 4 2 (- m – 0.5) = 4m² + 8m +4 D=0, 4m² + 8m +4 = 0 m² + 2m +1 = 0 (m + 1)² = 0 m= - 1 Substitute the found value of m into the original equation: 2x ² - 2x + 1 – 0.5 = 0 4x ² - 4x + 1 = 0 ( 2x – 1) ² =0 2x -1 =0 x = 0.5
Dynamic pause In the homework, 8th grade students were asked to find the roots of a quadratic trinomial (x² - 5x +7) ² - 2(x² - 5x +7) - 3 After thinking, Vitya reasoned this way: first you need to open the brackets, then bring similar terms . But Styopa said that there is a simpler way to solve it and it is not at all necessary to open the brackets. Help Vita find a rational solution
Dynamic pause Problems of finding the roots of a quadratic trinomial and composing quadratic equations are already found in ancient Egyptian mathematical papyri. The general rule for finding roots and solving equations of the form: ax ² + bx = c, where a > 0, b and c are any, was formulated by Brahmagupta (7th century AD). Brahmagupta did not yet know that a quadratic equation can also have a negative root. Bhaskara Acharya (12th century) formulated the relationships between the coefficients of the equation. Made a lot of problems.
Generalization, homework ◊ 1 Solving exercises with a parameter: various types of tasks; ◊ 2 Summary of the topic being studied; ◊ 3 Homework: by level. Homework
Generalization, homework Find the roots of the quadratic trinomial (x-4)² +(4y-12)². Find the values of the parameter a for each of which the quadratic trinomial x²+ 4 x + 2ax+8a+1 has one solution. Homework assignment: p.3; Group 1: No. 45 (c, d), No. 49 (c, d); Group 2: a) find the value of parameter a at which the square trinomial x²-6x+2ax+4a has no solution; b) find the roots of the quadratic trinomial (2x-6)²+(3y-12)²
source of the template Natalia Vladimirovna Chernakova Teacher of chemistry and biology, State Educational Institution NPO Arkhangelsk Region “Vocational School No. 31” “http://pedsovet.su/”
Finding the roots of a quadratic trinomial
Goals: introduce the concept of a quadratic trinomial and its roots; develop the ability to find the roots of a quadratic trinomial.
Lesson progress
I. Organizational moment.
II. Oral work.
Which of the numbers: –2; –1; 1; 2 – are the roots of the equations?
a) 8 X+ 16 = 0; V) X 2 + 3X – 4 = 0;
b) 5 X 2 – 5 = 0; G) X 3 – 3X – 2 = 0.
III. Explanation of new material.
Explanation of new material should be carried out according to the following scheme:
1) Introduce the concept of the root of a polynomial.
2) Introduce the concept of a quadratic trinomial and its roots.
3) Analyze the question of the possible number of roots of a square trinomial.
The question of isolating the square of a binomial from a square trinomial is best discussed in the next lesson.
At each stage of explaining new material, it is necessary to offer students an oral task to test their understanding of the main points of the theory.
Task 1. Which of the numbers: –1; 1; ; 0 – are the roots of the polynomial X 4 + 2X 2 – 3?
Assignment 2. Which of the following polynomials are quadratic trinomials?
1) 2X 2 + 5X – 1; 6) X 2 – X – ;
2) 2X – ; 7) 3 – 4X + X 2 ;
3) 4X 2 + 2X + X 3 ; 8) X + 4X 2 ;
4) 3X 2 – ; 9) 
+ 3X – 6;
5) 5X 2 – 3X; 10) 7X 2 .
Which quadratic trinomials have root 0?
Task 3. Can a square trinomial have three roots? Why? How many roots does a square trinomial have? X 2 + X – 5?
IV. Formation of skills and abilities.
Exercises:
1. № 55, № 56, № 58.
2. No. 59 (a, c, d), No. 60 (a, c).
In this task you do not need to look for the roots of quadratic trinomials. It is enough to find their discriminant and answer the question posed.
a) 5 X 2 – 8X + 3 = 0;
D 1 = 16 – 15 = 1;
D 1 0, which means that this quadratic trinomial has two roots.
b) 9 X 2 + 6X + 1 = 0;
D 1 = 9 – 9 = 0;
D 1 = 0, which means the square trinomial has one root.
c) –7 X 2 + 6X – 2 = 0;
7X 2 – 6X + 2 = 0;
D 1 = 9 – 14 = –5;
If there is time left, you can do No. 63.
Solution
Let ax 2 + bx + c is a given quadratic trinomial. Because a+ b +
+ c= 0, then one of the roots of this trinomial is equal to 1. By Vieta’s theorem, the second root is equal to . According to the condition, With = 4A, so the second root of this quadratic trinomial is equal to 
.
ANSWER: 1 and 4.
V. Lesson summary.
Frequently asked questions:
– What is the root of a polynomial?
– Which polynomial is called a quadratic trinomial?
– How to find the roots of a quadratic trinomial?
– What is the discriminant of a quadratic trinomial?
– How many roots can a square trinomial have? What does this depend on?
Homework: No. 57, No. 59 (b, d, f), No. 60 (b, d), No. 62.
Expanding polynomials to obtain a product can sometimes seem confusing. But it's not that difficult if you understand the process step by step. The article describes in detail how to factor a quadratic trinomial.
Many people do not understand how to factor a square trinomial, and why this is done. At first it may seem like a futile exercise. But in mathematics nothing is done for nothing. The transformation is needed to simplify the expression and make it easier to calculate.
A polynomial of the form – ax²+bx+c, called a quadratic trinomial. The term "a" must be negative or positive. In practice, this expression is called a quadratic equation. Therefore, sometimes they say it differently: how to expand a quadratic equation.
Interesting! A polynomial is called a square because of its largest degree, the square. And a trinomial - because of the 3 components.
Some other types of polynomials:
- linear binomial (6x+8);
- cubic quadrinomial (x³+4x²-2x+9).
Factoring a quadratic trinomial
First, the expression is equal to zero, then you need to find the values of the roots x1 and x2. There may be no roots, there may be one or two roots. The presence of roots is determined by the discriminant. You need to know its formula by heart: D=b²-4ac.
If the result D is negative, there are no roots. If positive, there are two roots. If the result is zero, the root is one. The roots are also calculated using the formula.
If, when calculating the discriminant, the result is zero, you can use any of the formulas. In practice, the formula is simply shortened: -b / 2a.
The formulas for different discriminant values are different.
If D is positive:
If D is zero:
Online calculators
There is an online calculator on the Internet. It can be used to perform factorization. Some resources provide the opportunity to view the solution step by step. Such services help to better understand the topic, but you need to try to understand it well.
Useful video: Factoring a quadratic trinomial
Examples
We suggest looking at simple examples of how to factor a quadratic equation.
Example 1
This clearly shows that the result is two x's because D is positive. They need to be substituted into the formula. If the roots turn out to be negative, the sign in the formula changes to the opposite.
We know the formula for factoring a quadratic trinomial: a(x-x1)(x-x2). We put the values in brackets: (x+3)(x+2/3). There is no number before a term in a power. This means that there is one there, it goes down.
Example 2
This example clearly shows how to solve an equation that has one root.
We substitute the resulting value:
Example 3
Given: 5x²+3x+7
First, let's calculate the discriminant, as in previous cases.
D=9-4*5*7=9-140= -131.
The discriminant is negative, which means there are no roots.
After receiving the result, you should open the brackets and check the result. The original trinomial should appear.
Alternative solution
Some people were never able to make friends with the discriminator. There is another way to factorize a quadratic trinomial. For convenience, the method is shown with an example.
Given: x²+3x-10
We know that we should get 2 brackets: (_)(_). When the expression looks like this: x²+bx+c, at the beginning of each bracket we put x: (x_)(x_). The remaining two numbers are the product that gives “c”, i.e. in this case -10. The only way to find out what numbers these are is by selection. The substituted numbers must correspond to the remaining term.
For example, multiplying the following numbers gives -10:
- -1, 10;
- -10, 1;
- -5, 2;
- -2, 5.
- (x-1)(x+10) = x2+10x-x-10 = x2+9x-10. No.
- (x-10)(x+1) = x2+x-10x-10 = x2-9x-10. No.
- (x-5)(x+2) = x2+2x-5x-10 = x2-3x-10. No.
- (x-2)(x+5) = x2+5x-2x-10 = x2+3x-10. Fits.
This means that the transformation of the expression x2+3x-10 looks like this: (x-2)(x+5).
Important! You should be careful not to confuse the signs.
Expansion of a complex trinomial
If “a” is greater than one, difficulties begin. But everything is not as difficult as it seems.
To factorize, you first need to see if anything can be factored out.
For example, given the expression: 3x²+9x-30. Here the number 3 is taken out of brackets:
3(x²+3x-10). The result is the already well-known trinomial. The answer looks like this: 3(x-2)(x+5)
How to decompose if the term that is in the square is negative? In this case, the number -1 is taken out of brackets. For example: -x²-10x-8. The expression will then look like this:
The scheme differs little from the previous one. There are just a few new things. Let's say the expression is given: 2x²+7x+3. The answer is also written in 2 brackets that need to be filled in (_)(_). In the 2nd bracket is written x, and in the 1st what is left. It looks like this: (2x_)(x_). Otherwise, the previous scheme is repeated.
The number 3 is given by the numbers:
- -1, -3;
- -3, -1;
- 3, 1;
- 1, 3.
We solve equations by substituting these numbers. The last option is suitable. This means that the transformation of the expression 2x²+7x+3 looks like this: (2x+1)(x+3).
Other cases
It is not always possible to convert an expression. With the second method, solving the equation is not required. But the possibility of transforming terms into a product is checked only through the discriminant.
It is worth practicing solving quadratic equations so that when using the formulas there are no difficulties.
Useful video: factoring a trinomial
Conclusion
You can use it in any way. But it’s better to practice both until they become automatic. Also, learning how to solve quadratic equations well and factor polynomials is necessary for those who are planning to connect their lives with mathematics. All the following mathematical topics are built on this.
