Question

Solve: $5^{(x+1)}+5^{(2-x)}=126$.

Answer 1

Given $5^{(x+1)}+5^{(2-x)}=126$
Applying the rule of $a^{m+n}=a^m\times a^n$
$\Rightarrow 5^x\times 5^1+5^2\times 5^{-x}=126$
Let $5^x=y\Rightarrow 5^{-x}=y^{-1}$.
So, above expression can be written as $5y+\frac{25}{y}=126$
$\Rightarrow \frac{5y^2+25}{y}=126$
$\Rightarrow 5y^2+25=126y$
$\Rightarrow 5y^2-126y+25=0$
$\Rightarrow 5y^2-y-125y+25=0$
$\Rightarrow y(5y-1)-25(5y-1)=0$
$\Rightarrow (5y-1)(y-25)=0$
$\Rightarrow 5y-1=0$ and $y-25=0$
$\therefore y=\frac{1}{5}$ and $y=25$
$\Rightarrow 5^x=5^{-1}$ and $5^x=5^2$ (since, $y=5^x$)
$\therefore x=-1$ and $2$.