Question

Решить уравнение
x⁴ - 8x³ + 15x² - 2x - 9 = 0

Answer 1

x⁴ - 8x³ + 15x² - 2x - 9 = 0
x⁴ - 8x³ + 24x² - 32x + 16 - 9x² + 30x - 25 = 0
(x⁴ - 4*x³*2 + 6*x²*2² - 4*x*2³ + 2⁴) - ( (3*x)² - 2*3x*5 + 5²) = 0
(x - 2)⁴ - (3x - 5)² = 0
( (x - 2)² - (3x - 5) )( (x - 2)² + (3x - 5) ) = 0
(x² - 4x + 4 - 3x + 5)(x² - 4x + 4 + 3x - 5) = 0
(x² - 7x + 9)(x² - x - 1) = 0

1) x² - 7x + 9 = 0
D = (-7)² - 4 * 9 = 49 - 36 = 13
$x_{1,2}=\frac{7\pm \sqrt{13}}{2}$

2) x² - x - 1 = 0
D = (-1)² - 4 * (-1) = 1 + 4 = 5
$x_{3,4}=\frac{1\pm \sqrt{5}}{2}$

Ответ: $\frac{7\pm \sqrt{13}}{2}; \frac{1\pm \sqrt{5}}{2}$