$\sqrt{x^2 + 3x - 4} + \sqrt{x^2 + 6x + 8} = 0$
$\sqrt{x^2 + 3x - 4}$ и $\sqrt{x^2 + 6x + 8} = 0$
а)
$x^2 + 3x - 4 = 0$
$D = b^2 - 4ac = 3^2 - 4 * 1 * (-4) = 9 + 16 = 25 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-3 + \sqrt{25}}{2 * 1} = \frac{-3 + 5}{2} = \frac{2}{2} = 1$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-3 - \sqrt{25}}{2 * 1} = \frac{-3 - 5}{2} = \frac{-8}{2} = -4$
б)
$x^2 + 6x + 8 = 0$
$D = b^2 - 4ac = 6^2 - 4 * 1 * 8 = 36 - 32 = 4 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-6 + \sqrt{4}}{2 * 1} = \frac{-6 + 2}{2} = \frac{-4}{2} = -2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-6 - \sqrt{4}}{2 * 1} = \frac{-6 - 2}{2} = \frac{-8}{2} = -4$
Ответ: −4

$x^2 - 4x + 4 + |x^2 - 3x + 2| = 0$
$(x - 2)^2 = 0$ и $|x^2 - 3x + 2| = 0$
а)
$(x - 2)^2 = 0$
x − 2 = 0
x = 2
б)
$x^2 - 3x + 2 = 0$
$D = b^2 - 4ac = (-3)^2 - 4 * 1 * 2 = 9 - 8 = 1 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{3 + \sqrt{1}}{2 * 1} = \frac{3 + 1}{2} = \frac{4}{2} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{3 - \sqrt{1}}{2 * 1} = \frac{3 - 1}{2} = \frac{2}{2} = 1$
Ответ: 2

$\sqrt{25 - x^2} + |x^2 + 8x - 20| = 0$
$\sqrt{25 - x^2} = 0$ и $|x^2 + 8x - 20| = 0$
а)
$25 - x^2 = 0$
(5 − x)(5 + x) = 0
5 − x = 0
x = 5
или
5 + x = 0
x = −5
б)
$x^2 + 8x - 20 = 0$
$D = b^2 - 4ac = 8^2 - 4 * 1 * (-20) = 64 + 80 = 144 > 0$
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{-8 + \sqrt{144}}{2 * 1} = \frac{-8 + 12}{2} = \frac{4}{2} = 2$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{-8 - \sqrt{144}}{2 * 1} = \frac{-8 - 12}{2} = \frac{-20}{2} = -10$
Ответ: нет корней