Question

In the following exercises,solve each equation.

$5^{x^2+2x}=\frac{1}{5}$.

Short answer

The solution is$x=-1$

Step-by-step solution

Step 1.Given information

Consider the given quadratic equation$5^{x^2+2x}=\frac{1}{5}$

Solve the equation for x as follows;

Step 2.First write both sides of the equation with the same base

Because left side of the equation has base 5,so write the right side of the equation with base 5 as

$5^{x^2+2x}=5^{-1}[\frac{1}{5}=5^{-1}]$

Because the base are same 5,so the exponent must be equal.

$x^2+2x=-1$

Step 3.Now solve the equation

$$\begin{gathered} x^2+2x+1=-1+1 \\ {x}^2+2x+1=0 \\ {(x+1)}^2=0x=-1 \\ So,thevalueofx=-1 \end{gathered}$$

Step 4.Check

To check the solution,replace x by -1 in original equation.

$$\begin{gathered} 5^{x^2+2x}=\frac{1}{5} \\ {5}^{{(-1)}^2+2(-1)}=\frac{1}{5} \\ {5}^{-1}=\frac{1}{5} \\ \frac{1}{5}=\frac{1}{5}\left[true\right] \\ Therefore,thevalueofx=\frac{1}{5} \end{gathered}$$