The purpose of the lesson: To generalize and put into the system all the knowledge and skills that we possess. Form skills, solve square

Lesson presentation

"Solving quadratic equations"

Updating basic knowledge

1. What kind of equation is called square?

A quadratic equation is an equation of the form Oh 2 + in + s= 0, where x is a variable, a, in and With- some numbers, moreover a not equal to 0.

2. Which expression is a quadratic equation?

7x - x 2 + 5 = 0

3. Name the coefficients in the equations:

5x 2 + 4x + 1 = 0 x 2 + 5 = 0 - x 2 + x = 0

a = 1; v = 0; With = 5

a = -1; v = 1; With = 0

a = - 5 ; v = 4; With = 1

4. Make a quadratic equation if

a = 5, v = -3, With = -2.

5x 2 - 3x - 2 = 0

5. What quadratic equations are called incomplete quadratic equations?

If in a quadratic equation a x 2 + v x + With= 0 at least one of the coefficients v or With is equal to zero,

then such an equation is called an incomplete quadratic equation.

6. Name the types of incomplete quadratic equations.

1) a x 2 + With = 0

2) a x 2 + v x = 0

3) a x 2 = 0

7 what is the expression called v 2 – 4 ace ?

Discriminant

0 two roots in 2 - 4 ac = 0 one root in 2 - 4 ac has no roots 9. Write the formula for the roots of a general quadratic equation. "Width =" 640 "

8. What does it mean?

v 2 – 4 ace 0

two roots

v 2 – 4 ace = 0

one root

v 2 – 4 ace

has no roots

9.Write the formula for the roots of a general quadratic equation.

1. Which expression is a quadratic equation?

Option 1. Option 2.

a) 3x + 1 = 0 a) 5x 2 + x - 4 = 0

b) 5x + 4x 2 = 0 b) 4x - 3 = 0

c) 4x 2 + x - 1 c) x 2 - x - 12

2. Which of the numbers are the roots of the equation?

Option 1. Option 2.

x 2 + 3x + 2 = 0 x 2 - 6x + 8 = 0

a) -1 and - 2 a) - 4 and 2

b) 2 and -1 b) 4 and -2

c) -2 and 1 c) 4 and 2

0 for 𝐃 = 0 a) one a) one b) two b) two c) none c) none "width =" 640 "

3. Determine the signs of the roots of the equation without solving it:

Option 1. Option 2.

x 2 -14x + 21 = 0 x 2 - 2x - 35 = 0

a) (- and +) a) (+ and +)

b) (- and -) b) (- and +)

c) (+ and +) c) (- and -)

4. How many roots does the equation have a x 2 + v x + With = 0

Option 1. Option 2.

for 𝐃 0 for 𝐃 = 0

a) one a) one

b) two b) two

c) none c) none

5. Without solving the equation, determine how many roots it has:

Option 1. Option 2.

5x 2 - 6x + 2 = 0 x 2 + 10x + 9 = 0

a) one a) one

b) two b) two

c) none c) none

Mutual verification:

Option 1. Option 2.

Key to the assignment Evaluation criterion No mistakes - 5 points 1-2 mistakes - 4 points 3-4 mistakes - 3 points 5-6 mistakes - 2 points More than 6 mistakes - 0 points

The first quadratic equations appeared a long time ago. They were solved in Babylon around 2000 BC, and Europe seven years ago celebrated the 800th anniversary of the quadratic equations, because it was in 1202 that the Italian scientist Leonard Fibonacci laid out the formulas for the quadratic equation. And only in the 17th century, thanks to Newton, Descartes and other scientists, these formulas took on a modern form.

0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D "title =" (! LANG: Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D = в² - 4ac 3. If D> 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D" class="link_thumb"> 7 !} Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D = в² - 4ac 3. If D> 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D "> 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D"> 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D "title =" (! LANG: Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D = в² - 4ac 3. If D> 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D"> title="Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D = в² - 4ac 3. If D> 0, then the equation has two roots 4. If D = 0, then the equation has one root 5. If D"> !}

"Hurry, but don't be mistaken!" Key to the test Evaluation criterion 1-B 2-B No errors - 5 points 1 error - 4 points 3 errors - 2 points 2 errors - 1 point 4-5 errors - 0 points

Performance map F.I.Warm-upSlightly - think a little Theory Questions Solving equations Catch an errorTestTotal Evaluation criteria: points - "5" 9-14 points - "4" 5-8 points - "3"

What determines the number of roots of a quadratic equation? Answer: From the sign D. D = 0 D 0 1 root No roots two roots X = -b / 2 aX = (- b + D) / 2 a 0 1 root No roots two roots X = -b / 2 aX = (- b + D) / 2 a "> 0 1 root No roots two roots X = -b / 2 aX = (- b + D) / 2 a "> 0 1 root No roots two roots X = -b / 2 aX = (- in + D) / 2 a" title = "(! LANG: What determines the number of roots of a quadratic equation? Answer: From the sign of D. D = 0 D 0 1 root No roots two roots X = -b / 2 aX = (- b + D) / 2 a"> title="What determines the number of roots of a quadratic equation? Answer: From the sign D. D = 0 D 0 1 root No roots two roots X = -b / 2 aX = (- b + D) / 2 a"> !}

Exercise. Flasks contain liquids in which quadratic equations float. If D> 0, then a vapor is released from the flask, in which the roots of the equation are located. If D 0, then a pair is released from the flask, in which the roots of the equation are located. If D "> 0, then a vapor is released from the flask, in which the roots of the equation are located. If D"> 0, then a pair is released from the flask, in which the roots of the equation are located. If D "title =" (! LANG: Task. The flasks are filled with liquids in which quadratic equations float. If D> 0, then steam is released from the flask, in which the roots of the equation are located. If D"> title="Exercise. Flasks contain liquids in which quadratic equations float. If D> 0, then a vapor is released from the flask, in which the roots of the equation are located. If D"> !}

The treatise and its content The first book that has come down to us, in which the classification of quadratic equations and methods for their solution, as well as geometric proofs of these solutions, is given, is the treatise "Kitab al-jabr wal-muqabala" by Muhammad al-Khwarizmi. Mathematician Muhammad al-Khwarizmi explains how to solve equations of the form ax 2 = bx, ax 2 = c, ax 2 + c = bx, ax 2 + bx = c, bx + c = ax 2 (letters a, b, c denote only positive numbers) and only looks for positive roots.

Problem “The square and number 21 is equal to 10 roots. Find the root (meaning the root of the equation is X 2 + 21 = 10X). The author's decision sounds something like this: “Divide the number of roots in half - you get 5, multiply 5 by itself, subtract 21 from the product, there will be 4. Extract the root of 4 - you get 2. subtract 2 from 5 - you get 3, this will be the desired root ... Or add to 5, which gives 7, this is also his root.

Research: a) consider the reduced quadratic equation X 2 + 3X-10 = 0; rewrite it as X 2 -10 = -3X. Solution: 1) divide the number of roots in half: -3: 2 = -1.5 2) multiply (-1.5) by itself: -1.5 * (- 1.5) = 2.25 3) from the product subtract (-10): 2.25 - (- 10) = 2.25 + 10 = 12.25

4) extract the square root of 12.25: we get 3.5 5) subtract 3.5 from (-1.5): -1.5-3.5 = -5- this will be the desired root first 6) add 3, 5 to (-1.5): -1.5 + 3.5 = 2- this will be the desired root of the second. Let's check: With X 1 = -5 With X 2 = = = 0 0 = 0 (true) Answer: X 1 = -5, X 2 = 2.

Conclusion: Indeed, the above method for solving the reduced quadratic equation in the treatise by the mathematician Muhammad al-Khwarizmi only for positive numbers, is applicable for negative numbers too. Let's compose an algorithm for solving the reduced quadratic equations by the method of Muhammad al-Khwarizmi.

Algorithm for solving 1) We write the equation in the form: X 2 + c = bX 2) Divide the number of roots b by 2 3) Square the result of item 2 4) Subtract the free term from the result of item 3 5) Extract the square root of the result item 4 6) Subtract the result of item 5 from the result of item 5, we obtain the first root 7) Add the result of item 5 to the result of item 2, we obtain the second root

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Slide captions:

ALGEBRA, grade 8 Lesson topic: "Quadratic equations" If you hear that someone does not like mathematics, do not believe. One cannot help but love her - one can only not know her.

an equation of the form ax 2 + bx + c = 0, where x is a variable, a, b and c are some numbers, and a 0. DEFINITION: A quadratic equation is called

COMPLETE SQUARE EQUATIONS INCOMPLETE SQUARE EQUATIONS SQUARE EQUATIONS a ≠ 0, b ≠ 0, c ≠ 0 a ≠ 0, b = 0, c = 0 2x 2 + 5x-7 = 0 6x + x 2 -3 = 0 X 2 -8x -7 = 0 25-10x + x 2 = 0 3x 2 -2x = 0 2x + x 2 = 0 125 + 5x 2 = 0 49x 2 -81 = 0

Option 1 a) 6x 2 - x + 4 = 0 b) 12x - x 2 = 0 c) 8 + 5x 2 = 0 Option 2 a) x - 6x 2 = 0 b) - x + x 2 - 15 = 0 c ) - 9x 2 + 3 = 0 1 option a) a = 6, b = -1, c = 4; b) a = -1, c = 12, c = 0; c) a = 5, b = 0, c = 8; Option 2 a) a = -6, b = 1, c = 0; b) a = 1, c = -1, c = - 15; c) a = -9, b = 0, c = 3. Determine the coefficients of the quadratic equation:

SOLUTION OF INCOMPLETE SQUARE EQUATIONS в = 0 ax 2 + c = 0 c = 0 ax 2 + bx = 0 b, c = 0 ax 2 = 0 1. Transfer c to the right side of the equation. ax 2 = -c 2. Division of both sides of the equation by a. x 2 = -s / a 3.If -s / a> 0 -two solutions: x 1 = and x 2 = - If -s / a

SOLVE INCOMPLETE EQUATIONS: Option 1: Option 2: a) 2x + 3x 2 = 0 a) 3x 2 - 2x = 0 b) 3x 2 - 243 = 0 b) 125 - 5x 2 = 0 c) 6x 2 = -10x - 2x (5 - 3x). c) -12x - 6x (2 - 3x) = 18x 2

Check friend 1 option a) x (2 + 3x) = 0, x = 0 or 2 + 3x = 0, 3x = -2, x = -2/3. Answer: 0 and -2/3. b) 3x 2 = 243, x 2 = 243/3, x 2 = 81, x = -9, x = 9. Answer: -9 and 9.c) 6x 2 = - 10x -10x + 6x 2, 6x 2 + 10x + 10x - 6x 2 = 0, 20x = 0, x = 0. Answer: 0.2 option a) x (3x -2) = 0, x = 0 or 3x-2 = 0, 3x = 2, x = 2/3. Answer: 0 and 2/3. b) - 5 x 2 = - 125, x 2 = -125 / -5, x 2 = 25, x = - 5, x = 5. Answer: -5 and 5. c) - 12x -12x +18 x 2 - 18 x 2 = 0, - 24x = 0, x = 0. Answer: 0.

Dynamic pause a) 3x 2 - 5x - 2 = 0 b) 4x 2 - 4x + 1 = 0 c) x 2 - 2x +3 = 0 d) 6x 2 - x + 4 = 0 e) 12x - x 2 = 0 f) 8 + 5x 2 = 0 g) 5x 2 - 4x + 2 = 0 h) 4x 2 - 3x -1 = 0 i) x 2 - 6x + 9 = 0 k) x - 6x 2 = 0 l) - x + x 2 - 15 = 0 m) - 9x 2 + 3 = 0

Methods for solving complete quadratic equations Selecting a square of a binomial. Formula: D = b 2 - 4ac, x 1,2 = Vieta's theorem.

What determines the number of roots of a quadratic equation? Answer: From sign D - discriminant. D = 0 D 0 1 root No roots two roots Х = -в / 2а Х = (- в + √D) / 2а

Calculate the discriminant and determine the number of roots of the quadratic equation 1 option a) 3x 2 - 5x - 2 = 0 b) 4x 2 - 4x + 1 = 0 c) x 2 - 2x +3 = 0 2 option a) 5x 2 - 4x + 2 = 0 b) 4x 2 - 3x -1 = 0 c) x 2 - 6x + 9 = 0

Check your friend D = b 2 -4ac 1 option a) D = (-5) 2 - 4 * 3 * (- 2) = 49, 2 roots; b) D = (-4) 2 - 4 * 4 * 1 = 0, 1 root; c) D = (-2) 2 - 4 * 1 * 3 = -8, no roots 2 option a) D = (-4) 2 - 4 * 5 * 2 = -24, no roots; D = (-3) 2 - 4 * 4 * (- 1) = 25, 2 roots; D = (-6) 2 - 4 * 1 * 9 = 0.1 root

SOLVE EQUATIONS using the formula: Option 1: Option 2: 2x 2 + 5x -7 = 0 2x 2 + 5x -3 = 0

Check yourself 1 option 2x 2 + 5x -7 = 0, D = 5 2 - 4 * 2 * (-7) = 81 = 9 2, x = (-5 -9) / 2 * 2 = -14 / 4 = - 3.5, x = (- 5 +9) / 4 = 4/4 = 1. Answer: -3.5 and 1.2 option 2x 2 + 5x -3 = 0, D = 5 2 - 4 * 2 * (-3) = 49 = 7 2, x = (-5 -7) / 2 * 2 = -12 / 4 = -3, x = (-5 +7) / 4 = 2/4 = 0.5. Answer: -3 and 0.5.

Historical background: Quadratic equations are first encountered in the work of the Indian mathematician and astronomer Aryabhatta. Another Indian scientist Brahmagupta (VII century) outlined the general rule for solving quadratic equations, which practically coincides with the modern one. In ancient India, public competition for solving difficult problems was common. The tasks were often clothed in poetic form. ________________________________________________ Here is the task of Bhaskara: Frisky flock of monkeys, having eaten their fill, had fun. The eighth part of them in the square was amused in the clearing. And twelve began to jump over the vines, hanging. How many monkeys were there, you tell me, in this flock?

Solution of Bhaskara's problem: Let there were x monkeys, then it was amusing in the clearing - (x / 8) 2 and 12 jumped over the vines. Let's make the equation: (x / 8) 2 + 12 = x, x 2/64 + 12 - x = 0, / * 64 x 2 - 64x + 768 = 0, D = (-64) 2 -4 * 1 * 768 = 4096 - 3072 = 1024 = 32 2, 2 roots x = (64 -32) / 2 = 16, x = (64 + 32) / 2 = 48. Answer: 16 or 48 monkeys.