Question

Решите уравнение
_______________

Answer 1

$9^{x- \frac{1}{2} } -8*3^{x-1} +5=0$
$9^{x}*9^{- \frac{1}{2} } -8*3^{x}*3^{-1} +5=0$
$(3^{2})^{x}* \frac{1}{9^{ \frac{1}{2} } } -8*3^{x}* \frac{1}{3} +5=0$
$(3^{x})^{2}* \frac{1}{(3^{2})^{ \frac{1}{2} } } - \frac{8}{3} *3^{x} +5=0$
$(3^{x})^{2}* \frac{1}{3} - \frac{8}{3} *3^{x} +5=0$
Пусть $t = 3^{x}$
$t^{2}* \frac{1}{3} - \frac{8}{3} *t +5=0$
$t^{2} - 8t + 15=0$
$D = (-8)^{2}-4*1*15 = 4 = 2^{2}$
$t_{1,2} = \frac{8 \frac{+}{} 2}{2}$
$t_{1} = 5$
$t_{2} = 3$
$\left \{ {{ 3^{x} = 5 } \atop { 3^{x} = 3 }} \right.$ ⇔ $\left \{ {{ x=log_{3}(5) } \atop { x=1 }} \right.$
Ответ: $1; log_{3}(5)$