Question

Solve given equation and verify your result using graphing utility.

$e^{x^2}=e^{3x}.\frac{1}{e^2}$

Short answer

The solution set is $\{1,2\}$ and it is verified through graph.

Step-by-step solution

Step 1. Given information 

The given equation is :

$e^{x^2}=e^{3x}.\frac{1}{e^2}$

We have to solve for x and verify using graphing utility.

Step 2. Get a single expression with base e on both sides.

Using law of exponents , get a single expression with base e on right side .

$e^{3x}.\frac{1}{e^2}=e^{3x}.e^{-2}=e^{3x-2}$

Step 3. Comparing the expressions having same bases and solving for x . 

Since bases are same , therefore the exponential powers should also be the same.

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

Factorise above expression.

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

Step 4. Verification using graphing utility 

Graph

$$\begin{gathered} Y_1=e^{x^2} \\ {Y}_2=e^{3x-2} \end{gathered}$$

Determine the point of intersection.

The graph intersects at $x=1,x=2$

So the solution set is$\{1,2\}$.