Найдите корни квадратного трёхчлена: a) 10x² + 5x - 5; б) -2x² + 12x - 18; в) x² - 2x - 4; г) 12x² - 12.

a) $$10x^2 + 5x - 5 = 0$$

a = 10, b = 5, c = -5

$$D = 5^2 - 4 \cdot 10 \cdot (-5) = 25 + 200 = 225$$

$$x_1 = \frac{-5 + \sqrt{225}}{2 \cdot 10} = \frac{-5 + 15}{20} = \frac{10}{20} = \frac{1}{2} = 0,5$$

$$x_2 = \frac{-5 - \sqrt{225}}{2 \cdot 10} = \frac{-5 - 15}{20} = \frac{-20}{20} = -1$$

Ответ: x₁ = 0,5, x₂ = -1

б) $$-2x^2 + 12x - 18 = 0$$

a = -2, b = 12, c = -18

$$D = 12^2 - 4 \cdot (-2) \cdot (-18) = 144 - 144 = 0$$

$$x = \frac{-12 \pm \sqrt{0}}{2 \cdot (-2)} = \frac{-12}{-4} = 3$$

Ответ: x = 3

в) $$x^2 - 2x - 4 = 0$$

a = 1, b = -2, c = -4

$$D = (-2)^2 - 4 \cdot 1 \cdot (-4) = 4 + 16 = 20$$

$$x_1 = \frac{2 + \sqrt{20}}{2 \cdot 1} = \frac{2 + 2\sqrt{5}}{2} = 1 + \sqrt{5}$$

$$x_2 = \frac{2 - \sqrt{20}}{2 \cdot 1} = \frac{2 - 2\sqrt{5}}{2} = 1 - \sqrt{5}$$

Ответ: x₁ = 1 + √5, x₂ = 1 - √5

г) $$12x^2 - 12 = 0$$

$$12x^2 = 12$$

$$x^2 = 1$$

$$x_1 = 1$$

$$x_2 = -1$$

Ответ: x₁ = 1, x₂ = -1