Presentation on the topic of the formula of the roots of a quadratic equation. Presentation "Another formula for the roots of a quadratic equation"

Schoolchildren face the solution of quadratic equations in the seventh grade for the first time. Throughout the course of algebra they encounter them more than once. There are many different methods for solving quadratic equations and formulas for finding their roots. This is what the presentation “Another formula for the roots of a quadratic equation” is devoted to. Thanks to the training file, students can independently understand the given examples, which will help them cope with similar tasks in the future. It will also be very useful to demonstrate the presentation in parallel with the lesson. This will help you better understand the material.

slides 1-2 (Presentation topic "Another formula for the roots of a quadratic equation", example)

The first slide contains a quadratic equation, and below are the formulas for the roots of this equation. As you can see, a slightly different discriminant formula is used here. The fact is that with an even coefficient for an unknown to the first degree, you can use another discriminant formula.

The equation is solved in terms of these formulas. It can be seen that the solution uses already studied material, for example, the properties of rational fractions, some transformations over them. Also, to solve this equation, students must remember the arithmetic root, how to extract it with sufficiently large radical expressions.

slides 3-4 (examples)

The next slide shows another example of solving a quadratic equation. Before looking at the solution, the student can independently try to solve it. If he understood the previous example well, he can handle this one as well. As a result, solutions can be compared.

In order for students to get the hang of it, it is proposed to solve two more examples. Thanks to detailed explanations, in the future, students will not have difficulty with similar examples that will be found in homework or tests.

slides 5 (example)

The presentation has a logical and coherent structure. Both text and formulas are displayed in the optimal size, corresponding to the standards for this kind of manuals. The colors match the requirements too. There are no distracting applications that are mistakenly present in many EMUs. Thus, students will be able to concentrate as much as possible on the topic and examples.

The material will also be useful for homeworkers and students who study externally.

These presentations make it easy to create a lesson plan. You can use the examples given in the file to demonstrate them during the lesson.

I stage. Warm-up Remember what equations are called quadratic, how to determine the coefficients a, b, c (textbook p. 133). Perform orally: 1. Are the equations quadratic? a) 2x 2 - 5x - 2 = 0; b) x 5 + 2x 2 = 0; c) 2xy - 3 = 0; d) x 2 + 4x \u003d 0 2. Determine the coefficients of quadratic equations: a) 2x 2 - 3x - 7 \u003d 0; b) 5x = 0; c) x 2 + 4x = 0 Test yourself!

II stage. Studying a new topic Read the text carefully: Let a quadratic equation ax 2 + bx + c = 0 be given. The solution of this equation begins with determining its discriminant. The discriminant of the quadratic equation ax 2 + bx + c = 0 is called an expression of the form b 2 - 4ac. The discriminant is denoted by the letter D. Next

II stage. Learning a new topic Number of roots of a quadratic equation Theorem 1. If D

II stage. Studying a new topic Theorem 2. If D = 0, then the quadratic equation has one root, which is found by the formula x = -b / 2a. Example 2. Solve the equation 4x x + 25 = 0 Solution: a = 4, b=-20, c = 25, D= b 2 - 4ac= (-20) * 4 * 25 = = = 0. By Theorem 2, the equation has one root: x = -b / 2a, x = 20 / 2 * 4 = 2.5. Answer: 2.5. NextBack

0, then the quadratic equation has two roots, which are found by the formulas: - 4 * 3 * (-11) = = 64 + 132 = 1" title="(!LANG: Stage II. Learning a new topic Theorem 3. If D > 0, then the quadratic equation has two roots, which are found by the formulas: , Example 3. Solve the equation 3x2 + 8x - 11 = 0 Solution: a = 3, b = 8, c = -11, D= b 2 - 4ac= 82 - 4 * 3 * (-11) = = 64 + 132 = 1" class="link_thumb"> 8 !} II stage. Studying a new topic Theorem 3. If D > 0, then the quadratic equation has two roots, which are found by the formulas:, Example 3. Solve the equation 3x2 + 8x - 11 = 0 Solution: a = 3, b = 8, c = -11 , D= b 2 - 4ac= * 3 * (-11) = = = 196. By Theorem 3, the equation has two roots:, x1 = () / 6 = 1 x2 = () / 6 = Answer: 1,. NextBack 0, then the quadratic equation has two roots, which are found by the formulas: - 4 * 3 * (-11) = = 64 + 132 = 1 "> 0, then the quadratic equation has two roots, which are found by the formulas: Example 3. Solve the equation 3x2 + 8x - 11 = 0 Solution: a = 3 , b = 8, c = -11, D= b 2 - 4ac= 82 - 4 * 3 * (-11) = = 64 + 132 = 196. By Theorem 3, the equation has two roots:, x1 = (-8 + 14) / 6 = 1 x2 = (-8 - 14) / 6 = Answer: 1, NextBack "> 0, then the quadratic equation has two roots, which are found by the formulas:, Example 3. Solve the equation 3x2 + 8x - 11 = 0 Solution: a = 3, b = 8, c = -11, D= b 2 - 4ac= 82 - 4 * 3 * (-11) = = 64 + 132 = 1" title="(!LANG:II stage Studying a new topic Theorem 3. If D > 0, then the quadratic equation has two roots, which are found by the formulas: Example 3. Solve the equation 3x2 + 8x - 11 = 0 Solution: a = 3, b = 8, c = -11, D= b 2 - 4ac= 82 - 4 * 3 * (-11) = = 64 + 132 = 1"> title="II stage. Studying a new topic Theorem 3. If D > 0, then the quadratic equation has two roots, which are found by the formulas:, Example 3. Solve the equation 3x2 + 8x - 11 = 0 Solution: a = 3, b = 8, c = -11 , D= b 2 - 4ac= 82 - 4 * 3 * (-11) = = 64 + 132 = 1"> !}

Stage III Consolidation of the studied material Perform exercises 1-3 in your notebook. You can return to the second stage if you have any questions. After completing the exercises, check yourself and correct the mistakes. 1. Solve the equation: x 2 + 3x - 4 = 0 2. Solve the equation: x x + 25 = 0 3. Solve the equation: 2x 2 +3x + 10 = 0

The formula for the roots of a quadratic equation. Presentation Likizyuk M.I.

Aims and objectives of the lesson To develop the ability to apply quadratic equations to solve algebraic and geometric problems; to continue the formation of practical and theoretical skills and abilities on the topic “Quadricular Equations”; To promote the ability to analyze the conditions of tasks, the development of the ability to reason, the development of cognitive interest, the ability to see the connection between mathematics and the surrounding life; Cultivate attentiveness and culture of thinking, independence and mutual assistance.

1. Organizational moment. Setting goals and objectives of the lesson. 2. Phonetic charging. 3. Oral questioning. Verbal counting. 4. Learning new material. 5. Fixing. Solving examples. 6. Physical minute. 7. Generalization. 8. The result of the lesson 9. Homework. Lesson Plan

Speak correctly in class. Coefficient Root Discriminant Variable

Oral survey 1. Define a quadratic equation, give examples. 2. Name the coefficients a, b, c in the equations: 3 x 2 -5x+2=0; -5 x 2 +3x-7=0 , x 2 +2x=0 ; 4x 2 -5=0 3. Define the given quadratic equation, give examples. 4. Name the given quadratic equation, in which the second coefficient and free term are equal to -2 (3)

Mental count 370+230= 7.2:1000= :50= 0.6∙100000= ∙ 30= 1200:10000= +340= 0.125∙1000000= +14= 75:100000=

Definition of a quadratic equation. Def. 1. A quadratic equation is an equation of the form ax 2 + b x + c \u003d 0, where x is a variable, a, b and c are some numbers, and a  0. The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, b is the second coefficient, and c is the free term. FROM

Discriminant of a quadratic equation Def. 2. The discriminant of the quadratic equation ax 2 + b x + c \u003d 0 is the expression b 2 - 4ac. It is denoted by the letter D, i.e. D \u003d b 2 - 4ac. Three cases are possible: D  0 D  0 D  0

If D  0 In this case, the equation ax 2 + b x + c \u003d 0 has two real roots:

Tasks Solve the equation 2x² - 5x +2=0 Solve the equation 2x² - 3x +5=0 Solve the equation x² -2x +1=0

that is, x 1 \u003d 2 and x 2 \u003d 0.5 are the roots of the given equation. Here a = 2, b = -5, c = 2 . We have D \u003d b 2 - 4ac \u003d (-5) 2 - 4  2  2 \u003d 9. Since D > 0, the equation has two roots. Let's find them by the formula Solve the equation 2x 2 - 5x + 2 = 0

Solve the equation 2x 2 - 3x + 5 = 0 where a = 2, b = -3, c = 5. Let's find the discriminant D \u003d b 2 - 4ac \u003d \u003d (-3) 2 - 4 2 5 \u003d -31, because D

Solve the equation x 2 - 2 x + 1 = 0 Here a = 1 , b = - 2 , c = 1 . We get D \u003d b 2 - 4ac \u003d (-2) 2 - 4 1 1 \u003d 0, since D \u003d 0 We got one root x \u003d 1. To tasks

No. 2. a) At what values of x are the values of the polynomials: (1-3x) (x + 1) and (x-1) (x + 1)? B) At what values of x are the values of the polynomials: (2-x) (2x + 1) and (x-2) (x + 2)? No. 1. Solve the equations: a) x 2 + 7x-44 \u003d 0; b) 9y 2 +6y+1=0; c) –2 t 2 +8t+2=0; d) a + 3a 2 \u003d -11. e) x 2 -10x-39 \u003d 0; f) 4y 2 -4y+1=0; g) –3 t 2 -12 t+ 6 =0; 3) 4a 2 +5= a.

Answers № 1. A) x=-11, x=4 B) y=-1/3 C) t=2±√5 D) no solution E) x=-3, x=13 E) y=1/ 2 G) t=-2±√6 H) no solution No. 2 A) x=1/2, x=-1 B) x=2, x=-1C

Summary of the lesson. 1. What did you learn new in the lesson? 2. What is D equal to? 3. How many roots does the equation have if D>0 D

slide presentation

Slide text: The formula for the roots of a quadratic equation Zhuravleva Lyudmila Borisovna teacher of mathematics at the Moscow gymnasium No. 1503

Slide text: Do you want to learn how to solve quadratic equations? NOT REALLY

Slide text: Contents Definition of a quadratic equation Discriminant of a quadratic equation Formula of the roots of a quadratic equation Tasks Useful material Test Self-study

Slide text: Definition of a quadratic equation. Def. 1. A quadratic equation is an equation of the form ax2 + bx + c \u003d 0, where x is a variable, a, b and c are some numbers, and a 0. The numbers a, b and c are the coefficients of the quadratic equation. The number a is called the first coefficient, b is the second coefficient, and c is the free term.

Slide text: Quadratic discriminant Def. 2. The discriminant of the quadratic equation ax2 + bx + c = 0 is the expression b2 - 4ac. It is denoted by the letter D, i.e. D=b2-4ac. Three cases are possible: D 0 D 0 D 0

Slide text: If D 0 In this case, the equation ax2 + bx + c = 0 has two real roots:

Slide text: If D = 0 In this case, the equation ax2 + bx + c = 0 has one real root:

Slide #10

Slide text: If D 0 The equation ax2 + bx + c = 0 has no real roots.

Slide #11

Slide text: The formula for the roots of the quadratic equation Summarizing the cases considered, we obtain the formula for the roots of the quadratic equation ax2 + bx + c = 0. To the test

Slide #12

Slide text: Tasks Solve the equation 2x2- 5x + 2 = 0. Solve the equation 2x2- 3x + 5 = 0. Solve the equation x2- 2x + 1 = 0.

Slide #13

Slide text: Solve the equation 2x2- 5x + 2 = 0 Here a = 2, b = -5, c = 2. We have D = b2- 4ac = (-5)2- 4 2 2 = 9. Since D > 0 , then the equation has two roots. Let's find them by the formula that is, x1 = 2 and x2 = 0.5 - the roots of the given equation. To tasks

Slide #14

Slide text: 2x2- 5x + 2 = 0; x1=2, x2=0.5

Slide #15

Slide text: Solve the equation 2x2- 3x + 5 = 0 Here a = 2, b = -3, c = 5. Find the discriminant D = b2- 4ac= = (-3)2- 4 2 5 = -31, because D

Slide #16

Slide text: Solve the equation x2- 2x + 1 = 0 Here a = 1, b = -2, c = 1. We get D = b2- 4ac = (-2)2- 4 1 1= 0, because D= 0 Got one root x = 1. To tasks

Slide #17

Slide text: Useful material Definition of a quadratic equation Definition of a reduced quadratic equation Definition of a discriminant Formula of roots of a quadratic equation Coefficients of a quadratic equation

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Slide text: Definition of the reduced quadratic equation Def. 3. A reduced quadratic equation is a quadratic equation whose first coefficient is 1. x2 + bx + c \u003d 0

Slide #19

Slide text: Test 1. Calculate the discriminant of the equation x2-5x-6=0. 0 -6 1 25 -5 49 Next question

Slide #20

Slide text: 2. How many roots does the equation have if D< 0? Три корня Один корень Два корня Корней не имеет Следующий вопрос