Question

Solve for x : $3(81{)}^x+3=9^{x+1}+9^x$.

Answer 1

We have
$3(81{)}^x+3=9^{x+1}+9^x$
$\Rightarrow 3(9^2{)}^x+3=9^x\ast 9+9^x$
$\Rightarrow 3(9^2{)}^x+3=9^x(9+1)$
$\Rightarrow 3(9^2{)}^x+3=9^x(10)$
$\Rightarrow 3(9^x{)}^2+3=9^x(10)$

Let $9^x=t$, then we have
$3t^2-10t+3=0$
$\Rightarrow 3t^2-9t-t+3=0$
$\Rightarrow 3t(t-3)-1(t-3)=0$
$\Rightarrow (3t-1)(t-3)=0$
$\Rightarrow t=\frac{1}{3},t=3$
$\Rightarrow 9^x=\frac{1}{3},3$
$\Rightarrow 3^{2x}=3^{-1},3^1$
$or2x=-1,2x=1$
$\Rightarrow x=\frac{-1}{2},\frac{1}{2}$