28.19 a) (x-2)² = 3x-8; 6) (3x-1) (x + 3) + 1 = x(1+ 6x); в) 5(x + 2)² = -6x+44; г) (x + 4) (2x - 1) = x(3x + 11).

a) (x-2)² = 3x-8

x² - 4x + 4 = 3x - 8

x² - 7x + 12 = 0

a = 1, b = -7, c = 12

$$D = b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot 12 = 49 - 48 = 1$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{7 + \sqrt{1}}{2 \cdot 1} = \frac{7 + 1}{2} = \frac{8}{2} = 4$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{7 - \sqrt{1}}{2 \cdot 1} = \frac{7 - 1}{2} = \frac{6}{2} = 3$$

б) (3x-1) (x + 3) + 1 = x(1+ 6x)

3x² + 9x - x - 3 + 1 = x + 6x²

3x² + 8x - 2 = x + 6x²

3x² - 7x + 2 = 0

a = 3, b = -7, c = 2

$$D = b^2 - 4ac = (-7)^2 - 4 \cdot 3 \cdot 2 = 49 - 24 = 25$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{7 + \sqrt{25}}{2 \cdot 3} = \frac{7 + 5}{6} = \frac{12}{6} = 2$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{7 - \sqrt{25}}{2 \cdot 3} = \frac{7 - 5}{6} = \frac{2}{6} = \frac{1}{3}$$

в) 5(x + 2)² = -6x+44

5(x² + 4x + 4) = -6x + 44

5x² + 20x + 20 = -6x + 44

5x² + 26x - 24 = 0

a = 5, b = 26, c = -24

$$D = b^2 - 4ac = 26^2 - 4 \cdot 5 \cdot (-24) = 676 + 480 = 1156$$

$$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-26 + \sqrt{1156}}{2 \cdot 5} = \frac{-26 + 34}{10} = \frac{8}{10} = \frac{4}{5}$$

$$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-26 - \sqrt{1156}}{2 \cdot 5} = \frac{-26 - 34}{10} = \frac{-60}{10} = -6$$

г) (x + 4) (2x - 1) = x(3x + 11)

2x² - x + 8x - 4 = 3x² + 11x

2x² + 7x - 4 = 3x² + 11x

x² + 4x + 4 = 0

a = 1, b = 4, c = 4

$$D = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 4 = 16 - 16 = 0$$

Так как дискриминант равен нулю, уравнение имеет один корень:

$$x = \frac{-b}{2a} = \frac{-4}{2 \cdot 1} = -2$$

Ответ: a) x₁ = 4, x₂ = 3; б) x₁ = 2, x₂ = 1/3; в) x₁ = 4/5, x₂ = -6; г) x = -2