Question

1)Решите неполные квадратные уравнения:1.24х-х²=02.81х²=1002)Решите уравнения:3.(х+4)²=3х+403)Решительно уравнения, применяя теорему, обратную т.Виета. х²-16х-63=04)Решительно задачу.Периметр прямоугольника 28 см. Найти его стороны, если площадь прямоугольника равна 33 см²5)Один из корней уравнениями х²+10х+р=0 равен -12. Найти другой корень и р.

Answer 1

1) 24x - x² = 0
x(24 - x) = 0
x₁ = 0            24 - x = 0
                     x₂ = 24

2) 81x² = 100
81x² - 100 = 0
(9x - 10)(9x + 10) = 0
9x - 10 = 0                         9x + 10 = 0
9x = 10                              9x = - 10
x₁ = 10/9 = 1 1/9                x₂ = - 10/9 = - 1 1/9

3) (x + 4)² = 3x + 40
x² + 8x + 16 - 3x - 40 = 0
x² + 5x - 24 = 0
D = 5² - 4 * 1 * (- 24) = 25 + 96 = 121 = 11²
$X _{1}= \frac{-5+11}{2}= \frac{6}{2}=3\\X _{2}= \frac{-5-11}{2}= \frac{-16}{2}=-8$

4) x² - 16x - 63 = 0
x₁ * x₂ = - 63
x₁ + x₂ = 16
Вы неверно записали уравнение

5) Если периметр равен  28 см, то полупериметр равен 14 см, а это означает, что если обозначить одну  сторону прямоугольника через x, то длина другой  стороны будет равна 14 - x . Площадь прямоугольника равна произведению двух сторон . Составим и решим уравнение:
x(14 - x) = 33
14x - x² - 33 = 0
x² - 14x + 33 = 0
x₁ = 11      x₂ = 3 корни найдены по теореме, обратной теореме Виетта.
Ответ: стороны прямоугольника 11 см и  3 см


6) x² + 10x + p = 0
x₁ = - 12
x₁ + x₂ = - 10
x₂ = - 10 - x₁ = - 10 - ( - 12) = - 10 + 12 = 2
p = x₁ * x₂ = - 12 * 2 = - 24

Answer 2

1) Решение неполных квадратных уравнений: a) 24x - x^2 = 0 To solve this equation, we can rewrite it as x^2 - 24x = 0 and factor out an x: x(x - 24) = 0

So the solutions are x = 0 and x = 24.

b) 81x^2 = 100 To solve this equation, we can take the square root of both sides: 9x = ±10

Dividing both sides by 9, we get: x = ±10/9

2) Решение уравнений: a) (x + 4)^2 = 3x + 40 Expanding the left side: x^2 + 8x + 16 = 3x + 40

Rearranging the terms: x^2 + 5x - 24 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula: (x + 8)(x - 3) = 0

So the solutions are x = -8 and x = 3.

3) Применение теоремы, обратной к теореме Виета: a) x^2 - 16x - 63 = 0 According to the theorem, the sum of the roots of a quadratic equation is equal to the coefficient of the linear term with the opposite sign, divided by the coefficient of the quadratic term. In this case, the sum of the roots is -16.

The product of the roots is equal to the constant term divided by the coefficient of the quadratic term. In this case, the product of the roots is -63.

Therefore, we can write two equations: -16 = x1 + x2 -63 = x1 * x2

From the first equation, we can express x1 in terms of x2: x1 = -16 - x2

Substituting this into the second equation, we get: -63 = (-16 - x2) * x2

Simplifying and rearranging the terms, we obtain a quadratic equation: x2^2 + 16x2 - 63 = 0

Now we can solve this equation to find the values of x2. Once we have x2, we can substitute it back into x1 = -16 - x2 to find the value of x1.

4) Решение задачи о прямоугольнике: Given that the perimeter of the rectangle is 28 cm and the area is 33 cm^2, we can set up two equations based on the definitions of perimeter and area.

Let's assume the length of the rectangle is L and the width is W.

The perimeter equation is: 2L + 2W = 28

The area equation is: L * W = 33

We have a system of two equations. We can solve it using various methods, such as substitution or elimination.

Let's solve it by substitution. From the perimeter equation, we can express L in terms of W: L = 14 - W

Substituting this into the area equation: (14 - W) * W = 33

Simplifying, we get: W^2 - 14W + 33 = 0

Now we can solve this quadratic equation to find the values of W. Once we have W, we can substitute it back into L = 14 - W to find the value of L.

5) Нахождение другого корня и p: Given that one of the roots of the equation x^2 + 10x + p = 0 is -12, we can use the fact that the sum of the roots of a quadratic equation is equal to the opposite of the coefficient of the linear term, divided by the coefficient of the quadratic term.

The sum of the roots is -10 (since the coefficient of the linear term is 10). We know one root is -12, so the other root is: -10 - (-12) = 2

Therefore, the other root is 2.

To find the value of p, we can use the fact that the product of the roots is equal to the constant term divided by the coefficient of the quadratic term. In this case, the constant term is p.

The product of the roots is -12 * 2 = -24. So we have: -24 = p

Therefore, the value of p is -24.