Answer
solution set is $\left\{1, 2\right\}$.
Work Step by Step
Simplify the right side of the equation to obtain:
$e^{x^2} = \dfrac{e^{3x}}{e^2}$
Use the rule $\dfrac{a^m}{a^n} = a^{m-n}$ to obtain:
$e^{x^2} =e^{3x-2}$
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
$x^2=3x-2$
Subtract $3x$ and add $2$ on both sides of the equation to obtain:
$\begin{array}{ccc}
&x^2-3x+2 &= &3x-2-3x+2
\\&x^2-3x+2 &= &0\end{array}$
Factor the trinomial to obtain:
$(x-2)(x-1)=0$
Equate each factor to zero, and then solve each equation to obtain:
$\begin{array}{ccc}
&x-2=0 &\text{ or } &x-1=0
\\&x=2 &\text{ or } &x=1\end{array}$
Thus, the solution set is $\left\{1, 2\right\}$.
