Question

Решите легкое уравнение со степенями

Answer 1

$2 times {4}^{3 {x}^{2} + 5x } - 4 = 7 times {8}^{ {x}^{2} + frac{5x}{3} } \ \ 2 times {4}^{3 {x}^{2} + 5x } - 4 - 7 times {2}^{3 {x}^{2} + 5x } = 0$

${2}^{3 {x}^{2} + 5x} = t$

t > 0


$2 {t}^{2} - 4 - 7t = 0 \ 2 {t}^{2} - 7t - 4 = 0$

D = 49 - 4•2•(-4) = 49 + 32 = 81


$t1 = frac{ 7 - 9}{4} = frac{ - 2}{4} = - frac{1}{2} \ \ t2 = frac{7 + 9}{4} = frac{16}{4} = 4$


t = - 1/2 - не подходит, t > 0


Обратная замена:


$t = 4 \ \ {2}^{3 {x}^{2} + 5x } = {2}^{2} \ \ 3 {x}^{2} + 5x - 2 = 0$

D = 25 - 4•3•(-2) = 25 + 24 = 49


$x1 = frac{ - 5 - 7}{6} = frac{ - 12}{6} = - 2 \ \ x2 = frac{ - 5 + 7}{6} = frac{2}{6} = frac{1}{3}$


ОТВЕТ: - 2 ; 1/3