Question

Определите самый большой корень уравнения:

Answer 1

Ответ:

$\boxed{x = 3}$

Объяснение:

$2(\log_{9}{x})^{2} - 2 + 3 \log_{9}{x} = 0$

ОДЗ: x > 0 ⇒ x ∈ (0;+∞)

Замена: $t = \log_{9}{x}$

$2t^{2} - 2 + 3t =0$

$2t^{2} + 3t - 2 = 0$

$D = 9 - 4 * 2 * (-2) = 9 + 16 = 25 = 5^{2}$

$t_{1} = \dfrac{-3 + 5}{2 * 2} = \dfrac{2}{2 * 2} = \dfrac{1}{2}$

$t_{2} = \dfrac{-3 - 5}{2 * 2} = \dfrac{-8}{4} = -2$

$\left[ \begin{gathered} \log_{9}{x} = t_{1} \\ \log_{9}{x} = t_{2} \end{gathered} \right.$  $\left[ \begin{gathered} \log_{9}{x} =\dfrac{1}{2} \\ \log_{9}{x} =-2 \end{gathered} \right.$ $\left[ \begin{gathered} x = 3 \\ x = \dfrac{1}{81} \end{gathered} \right.$

Наибольший x = 3