Question

In the following exercises, solve each equation.

$\frac{e^{x^2}}{e^3}=e^{2x}$

Short answer

The answer is$x=-1,x=3$

Step-by-step solution

Step 1. Given

The equation is$\frac{e^{x^2}}{e^3}=e^{2x}$

Step 2. Solve the equation

$$\begin{gathered} \frac{e^{x^2}}{e^3}=e^{2x} \\ {e}^{x^2-3}=e^{2x} \end{gathered}$$

Bases are same so the exponents should be equal.

$$\begin{gathered} x^2-3=2x \\ {x}^2-2x-3=0 \\ {x}^2-3x+x-3=0 \\ x(x-3)+1(x-3)=0 \\ (x-3)(x+1)=0 \end{gathered}$$

So,

$$\begin{gathered} x=3 \\ x=-1 \end{gathered}$$

Step 3. Check the answers

Put $x=3$in given equation,

$$\begin{gathered} \frac{e^{x^2}}{e^3}=e^{2x} \\ \frac{e^{{\left(3\right)}^2}}{e^3}=e^{2\left(3\right)} \\ \frac{e^9}{e^3}=e^6 \\ {e}^{9-3}=e^6 \\ {e}^6=e^6 \end{gathered}$$

True

Put $x=-1$in given equation,

$$\begin{gathered} \frac{e^{x^2}}{e^3}=e^{2x} \\ \frac{e^{{\left(-1\right)}^2}}{e^3}=e^{2\left(-1\right)} \\ \frac{e^1}{e^3}=e^{-2} \\ {e}^{1-3}=e^{-2} \\ {e}^{-2}=e^{-2} \end{gathered}$$

True