Question

Solve The Equation $5^{2x}-24\times 5^x-25=0$ Find The Roots.

Answer 1

The given equation is $5^{2x}-24\times 5^x-25=0$

Let $5^x=t$

The given equation gets reduced to, $t^2-24t-25=0$

Using factorization method,

$t^2-25t+t-25=0$

$t(t-25)+1(t-25)=0$

$\left(t+1\right)\left(t-25\right)=0$

$$\begin{gathered} \left(t+1\right)=0or\left(t-25\right)=0 \\ t=-1ort=25 \end{gathered}$$

Neglecting $t=-1$

$t=25\Rightarrow 5^x=25\Rightarrow 5^x=5^2\Rightarrow x=2$

Therefore $x=2$ is the only root for the given equation.