Explanation

(b)
$\frac{3}{4}\left( \log_{2}x \right)^{2} + \log_{2}{x - \frac{5}{4} = \log_{x}\sqrt{2}}$
$\Rightarrow \ \ \frac{3}{4}\left( \log_{2}x \right)^{2} + \log_{2}{x - \frac{5}{4}} = \frac{1}{2\log_{2}x}$
$\Rightarrow \ \ 3\left( \log_{2}x \right)^{3} + 4\left( \log_{2}x \right)^{2} - 5\left( \log_{2}x \right) - 2 = 0$
Put $\log_{2}{x = y}$
$\therefore\ \ \ 3y^{2} + 4y^{2} - 5y - 2 = 0$
$\Rightarrow \ \ \ (y - 1)(y + 2)(3y + 1) = 0$
$\Rightarrow \ \ \ y = 1,\ - 2,\ \frac{- 1}{3}$
$\Rightarrow \ \ \log_{2}{x = 1,\ - 2,\ - \frac{1}{3}}$
$\Rightarrow \ \ x = 2,\frac{1}{2^{1/3}},\frac{1}{4}$