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 \u003d 0, where x is a variable, a, in and from - some numbers, and and not equal to 0.

2. Which expression is a quadratic equation?

7x - x 2 + 5 \u003d 0

3. Name the coefficients in the equations:

5x 2 + 4x + 1 \u003d 0 x 2 + 5 \u003d 0 - x 2 + x \u003d 0

and = 1; at = 0; from = 5

and = -1; at = 1; from = 0

and = - 5 ; at = 4; from = 1

4. Make a quadratic equation if

and = 5, at = -3, from = -2.

5x 2 - 3x - 2 \u003d 0

5. What quadratic equations are called incomplete quadratic equations?

If in a quadratic equation and x 2 + at x + from \u003d 0 at least one of the coefficients at or from 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 + from = 0

2) a x 2 + at x \u003d 0

3) a x 2 \u003d 0

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

Discriminant

0 two roots in 2 - 4 ac \u003d 0 one root in 2 - 4 ac has no roots 9. Write the formula for the roots of a general quadratic equation. "Width \u003d" 640 "

8. What does it mean?

at 2 – 4 ace 0

two roots

at 2 – 4 ace = 0

one root

at 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 \u003d 0 a) 5x 2 + x - 4 \u003d 0

b) 5x + 4x 2 \u003d 0 b) 4x - 3 \u003d 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 \u003d 0 x 2 - 6x + 8 \u003d 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 \u003d 0 a) one a) one b) two b) two c) none c) none "width \u003d" 640 "

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

Option 1. Option 2.

x 2 -14x + 21 \u003d 0 x 2 - 2x - 35 \u003d 0

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

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

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

4. How many roots does the equation have and x 2 + at x + from = 0

Option 1. Option 2.

for 𝐃 0 for 𝐃 \u003d 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 \u003d 0 x 2 + 10x + 9 \u003d 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 errors - 5 points 1-2 errors - 4 points 3-4 errors - 3 points 5-6 errors - 2 points More than 6 errors - 0 points

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

0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D "title \u003d" (! LANG: Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 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 \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D "\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D"\u003e 0, then the equation has two roots 4. If D \u003d 0, then the equation has one root 5. If D "title \u003d" (! LANG: Algorithm for solving a quadratic equation 1. Find the coefficients of the equation 2. Calculate the discriminant by the formula D \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 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 \u003d в² - 4ac 3. If D\u003e 0, then the equation has two roots 4. If D \u003d 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 the 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 \u003d 0 D 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a "\u003e 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a "\u003e 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- in + D) / 2 a" title \u003d "(! LANG: What determines the number of roots of a quadratic equation? Answer: From the sign D. D \u003d 0 D 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a"> title="What determines the number of roots of a quadratic equation? Answer: From the sign D. D \u003d 0 D 0 1 root No roots two roots X \u003d -b / 2 aX \u003d (- b + D) / 2 a"> !}

The task. The flasks are filled with liquids in which quadratic equations float. If D\u003e 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 "\u003e 0, then a vapor is released from the flask, in which the roots of the equation are located. If D"\u003e 0, then a pair is released from the flask, in which the roots of the equation are located. If D "title \u003d" (! LANG: Task. The flasks are filled with liquids in which quadratic equations float. If D\u003e 0, then steam is released from the flask, in which the roots of the equation are located. If D"> title="The task. The flasks are filled with liquids in which quadratic equations float. If D\u003e 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 are given, as well as geometric proofs of these solutions, 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 \u003d bx, ax 2 \u003d c, ax 2 + c \u003d bx, ax 2 + bx \u003d c, bx + c \u003d ax 2 (letters a, b, c denote only positive numbers) and only finds 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 \u003d 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 from 4 - you get 2. subtract 2 from 5 - you get 3, this will be the desired root ... Or add 5 to 7, which is also his root.

Research: a) consider the given quadratic equation X 2 + 3X-10 \u003d 0; rewrite it as X 2 -10 \u003d -3X. Solution: 1) divide the number of roots in half: -3: 2 \u003d -1.5 2) multiply (-1.5) by itself: -1.5 * (- 1.5) \u003d 2.25 3) from the product subtract (-10): 2.25 - (- 10) \u003d 2.25 + 10 \u003d 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 \u003d -5- this will be the desired root first 6) add 3, 5 to (-1.5): -1.5 + 3.5 \u003d 2- this will be the desired root of the second. Let's check: With X 1 \u003d -5 With X 2 \u003d \u003d \u003d 0 0 \u003d 0 (true) Answer: X 1 \u003d -5, X 2 \u003d 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.

Solution algorithm 1) Write the equation in the form: X 2 + c \u003d 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 to obtain the first root 7) Add the result of item 5 to the result of item 2 to 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 \u003d 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 \u003d 0, c \u003d 0 2x 2 + 5x-7 \u003d 0 6x + x 2 -3 \u003d 0 X 2 -8x -7 \u003d 0 25-10x + x 2 \u003d 0 3x 2 -2x \u003d 0 2x + x 2 \u003d 0 125 + 5x 2 \u003d 0 49x 2 -81 \u003d 0

Option 1 a) 6x 2 - x + 4 \u003d 0 b) 12x - x 2 \u003d 0 c) 8 + 5x 2 \u003d 0 Option 2 a) x - 6x 2 \u003d 0 b) - x + x 2 - 15 \u003d 0 c ) - 9x 2 + 3 \u003d 0 1 option a) a \u003d 6, b \u003d -1, c \u003d 4; b) a \u003d -1, c \u003d 12, c \u003d 0; c) a \u003d 5, b \u003d 0, c \u003d 8; Option 2 a) a \u003d -6, b \u003d 1, c \u003d 0; b) a \u003d 1, c \u003d -1, c \u003d - 15; c) a \u003d -9, b \u003d 0, c \u003d 3. Determine the coefficients of the quadratic equation:

SOLUTION OF INCOMPLETE SQUARE EQUATIONS в \u003d 0 ax 2 + c \u003d 0 c \u003d 0 ax 2 + bx \u003d 0 b, c \u003d 0 ax 2 \u003d 0 1. Transfer c to the right side of the equation. ax 2 \u003d -c 2. Division of both sides of the equation by a. x 2 \u003d -s / a 3. If -s / a\u003e 0 -two solutions: x 1 \u003d and x 2 \u003d - If -s / a

SOLVE INCOMPLETE EQUATIONS: Option 1: Option 2: a) 2x + 3x 2 \u003d 0 a) 3x 2 - 2x \u003d 0 b) 3x 2 - 243 \u003d 0 b) 125 - 5x 2 \u003d 0 c) 6x 2 \u003d -10x - 2x (5 - 3x). c) -12x - 6x (2 - 3x) \u003d 18x 2

Check your friend option 1 a) x (2 + 3x) \u003d 0, x \u003d 0 or 2 + 3x \u003d 0, 3x \u003d -2, x \u003d -2/3. Answer: 0 and -2/3. b) 3x 2 \u003d 243, x 2 \u003d 243/3, x 2 \u003d 81, x \u003d -9, x \u003d 9. Answer: -9 and 9.c) 6x 2 \u003d - 10x -10x + 6x 2, 6x 2 + 10x + 10x - 6x 2 \u003d 0, 20x \u003d 0, x \u003d 0. Answer: 0.2 option a) x (3x -2) \u003d 0, x \u003d 0 or 3x-2 \u003d 0, 3x \u003d 2, x \u003d 2/3. Answer: 0 and 2/3. b) - 5 x 2 \u003d - 125, x 2 \u003d -125 / -5, x 2 \u003d 25, x \u003d - 5, x \u003d 5. Answer: -5 and 5. c) - 12x -12x +18 x 2 - 18 x 2 \u003d 0, - 24x \u003d 0, x \u003d 0. Answer: 0.

Dynamic pause a) 3x 2 - 5x - 2 \u003d 0 b) 4x 2 - 4x + 1 \u003d 0 c) x 2 - 2x +3 \u003d 0 d) 6x 2 - x + 4 \u003d 0 e) 12x - x 2 \u003d 0 f) 8 + 5x 2 \u003d 0 g) 5x 2 - 4x + 2 \u003d 0 h) 4x 2 - 3x -1 \u003d 0 i) x 2 - 6x + 9 \u003d 0 k) x - 6x 2 \u003d 0 l) - x + x 2 - 15 \u003d 0 m) - 9x 2 + 3 \u003d 0

Methods for solving complete quadratic equations Selecting a square of a binomial. Formula: D \u003d b 2 - 4ac, x 1,2 \u003d Vieta's theorem.

What determines the number of roots of a quadratic equation? Answer: From sign D - discriminant. D \u003d 0 D 0 1 root No roots two roots Х \u003d -в / 2а Х \u003d (- в + √D) / 2а

Calculate the discriminant and determine the number of roots of the quadratic equation option 1 a) 3x 2 - 5x - 2 \u003d 0 b) 4x 2 - 4x + 1 \u003d 0 c) x 2 - 2x +3 \u003d 0 2 option a) 5x 2 - 4x + 2 \u003d 0 b) 4x 2 - 3x -1 \u003d 0 c) x 2 - 6x + 9 \u003d 0

Check your friend D \u003d b 2 -4ac 1 option a) D \u003d (-5) 2 - 4 * 3 * (- 2) \u003d 49, 2 roots; b) D \u003d (-4) 2 - 4 * 4 * 1 \u003d 0, 1 root; c) D \u003d (-2) 2 - 4 * 1 * 3 \u003d -8, no roots 2 option a) D \u003d (-4) 2 - 4 * 5 * 2 \u003d -24, no roots; D \u003d (-3) 2 - 4 * 4 * (- 1) \u003d 25, 2 roots; D \u003d (-6) 2 - 4 * 1 * 9 \u003d 0, 1 root

SOLVE EQUATIONS using the formula: Option 1: Option 2: 2x 2 + 5x -7 \u003d 0 2x 2 + 5x -3 \u003d 0

Check yourself 1 option 2x 2 + 5x -7 \u003d 0, D \u003d 5 2 - 4 * 2 * (-7) \u003d 81 \u003d 9 2, x \u003d (-5 -9) / 2 * 2 \u003d -14 / 4 \u003d - 3.5, x \u003d (- 5 + 9) / 4 \u003d 4/4 \u003d 1. Answer: -3.5 and 1.2 option 2x 2 + 5x -3 \u003d 0, D \u003d 5 2 - 4 * 2 * (-3) \u003d 49 \u003d 7 2, x \u003d (-5 -7) / 2 * 2 \u003d -12 / 4 \u003d -3, x \u003d (-5 +7) / 4 \u003d 2/4 \u003d 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 difficult problems was common. The problems were often clothed in poetic form. ________________________________________________ Here is the problem of Bhaskara: Frisky flock of monkeys, having eaten their fill, had fun. The eighth part of them in the square amused themselves in the clearing. And twelve began to jump over the vines, hanging. How many monkeys were there, tell me, in this pack?

Solution to 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 \u003d x, x 2/64 + 12 - x \u003d 0, / * 64 x 2 - 64x + 768 \u003d 0, D \u003d (-64) 2 -4 * 1 * 768 \u003d 4096 - 3072 \u003d 1024 \u003d 32 2, 2 roots x \u003d (64 -32) / 2 \u003d 16, x \u003d (64 + 32) / 2 \u003d 48. Answer: 16 or 48 monkeys.