Solve $5^{x+1}+5^{2-x}=5^3+1$

The correct options are A: $x=2,-1$

$5^{x+1}+5^{2-x}=5^3+1$

$\Rightarrow 5^x\times 5+5^2\times 5^{-x}=125+1=126$

$\Rightarrow 5^x\times 5+\frac{25}{5^x}=\mathrm{126......}\left(i\right)$

Let $5^x=y$

Then eq. (i) reduces to,

$\Rightarrow 5y+\frac{25}{y}=126$

$\Rightarrow 5y^2-126y+25=0$

$\Rightarrow 5y^2-125y-y+25=0$

$\Rightarrow 5y(y-25)-1(y-25)=0$

$\Rightarrow (y-25)(5y-1)=0$

$\Rightarrow y-25=0or5y-1=0$

$\Rightarrow y=25ory=\frac{1}{5}$

$\Rightarrow 5^x=25or{5}^x=\frac{1}{5}$ [Putting $5^x=y$]

$\Rightarrow 5^x=5^{2}or{5}^x=5^{-1}$

$\Rightarrow x=2orx=-1$

$\therefore$ Roots of the equation are $2$ and $-1.$