Question

In the following exercises, solve each equation.

$\frac{e^{x^2}}{e^2}=e^x$

Short answer

The answer is$x=-1,x=2$

Step-by-step solution

Step 1. Given

The equation is$\frac{e^{x^2}}{e^2}=e^x$

Step 2. Solve the equation

$$\begin{gathered} \frac{e^{x^2}}{e^2}=e^x \\ {e}^{x^2-2}=e^x \end{gathered}$$

Bases are same so the exponents should be equal.

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

So,

$$\begin{gathered} x=-1 \\ x=2 \end{gathered}$$

Step 3. Check the answers

Put $x=-1$in given equation,

$$\begin{gathered} \frac{e^{x^2}}{e^2}=e^x \\ \frac{e^{{\left(-1\right)}^2}}{e^2}=e^{\left(-1\right)} \\ \frac{e^1}{e^2}=\frac{1}{e} \\ {e}^{1-2}=\frac{1}{e} \\ {e}^{-1}=\frac{1}{e} \\ \frac{1}{e}=\frac{1}{e} \end{gathered}$$

True

Put$x=2$in given equation,

$$\begin{gathered} \frac{e^{x^2}}{e^2}=e^x \\ \frac{e^{{\left(2\right)}^2}}{e^2}=e^2 \\ \frac{e^4}{e^2}=e^2 \\ {e}^{4-2}=e^2 \\ {e}^2=e^2 \end{gathered}$$

True