Solve the equation: $\;$ $15 \times 2^{x+1} + 15 \times 2^{-x+2} = 135$
Given equation: $\;\;$ $15 \times 2^{x+1} + 15 \times 2^{-x+2} = 135$
i.e. $\;$ $15 \times 2^x \times 2^1 + 15 \times 2^{-x} \times 2^2 = 135$
i.e. $\;$ $30 \times 2^x + \dfrac{60}{2^x} = 135$
i.e. $\;$ $2 \times 2^x + \dfrac{4}{2^x} = 9$ $\;\;\; \cdots \; (1)$
Let $\;$ $2^x = p$ $\;\;\; \cdots \; (2)$
Then equation $(1)$ becomes
$2p + \dfrac{4}{p} = 9$
i.e. $\;$ $2 p^2 - 9p + 4 = 0$
i.e. $\;$ $\left(p - 4\right) \left(2p - 1\right) = 0$
i.e. $\;$ $p = 4$ $\;$ or $\;$ $p = \dfrac{1}{2}$
When $\;$ $p = 4$, $\;$ we have from equation $(2)$,
$2^x = 4 = 2^2$ $\implies$ $x = 2$
When $\;$ $p = \dfrac{1}{2}$, $\;$ we have from equation $(2)$,
$2^x = \dfrac{1}{2} = 2^{-1}$ $\implies$ $x = -1$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{-1, \; 2 \right\}$