Question

Solve the following equation:
$4^{\frac{2}{x}}-{5.4}^{\frac{1}{x}}+4=0$

Answer 1

$4^{\frac{2}{x}}-{5.4}^{\frac{1}{x}}+4=0$

Let $\frac{1}{x}=t\Rightarrow x\ne 0$

$\Rightarrow 4^{2t}-{5.4}^t+4=0$

$\Rightarrow (4t{)}^2-5.(4^t)+4=0$

Let $4^t=u$

$\Rightarrow u^2-5u+4=0$

$\Rightarrow u^2-4u-u+4=0$

$\Rightarrow (u-4)(u-1)=0$

$\Rightarrow u=4,1$

$\Rightarrow 4^t=4,1$

$\Rightarrow 4^t=4$ & $4^t=1$

$\Rightarrow t=1$ & $t=0$

$\Rightarrow \frac{1}{x}=1,0$

$\Rightarrow x=1$