Question

Find x

$2^{2x+2}-6^x-2\times 3^{2x+2}=0$

Answer 1

The equation can be written as
${4.2}^{2x}-2^x\cdot 3^x-{18.3}^{2x}=0$
Dividing by $2^x\cdot 3^x$,we get
$4{\left(\frac{2}{3}\right)}^x-1-18{\left(\frac{2}{3}\right)}^x=0$
Put ${\left(\frac{2}{3}\right)}^x=t$. Then $\frac{4}{t}-1-18t=0$
or $18t^2+t-4=0$ or $(2t+1)(9t-4)=0$
$\therefore t=\frac{4}{9}or\frac{-1}{2}$
$\therefore t={\left(\frac{3}{2}\right)}^x=\left(\frac{4}{9}\right)={\left(\frac{2}{3}\right)}^2={\left(\frac{3}{2}\right)}^{-2}$
giving $x=-2$