Q. 79 - Precalculus Enhanced with Graphing Utilities

Question

Solve given equation and verify your result using graphing utility. 

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

Step-by-Step Solution

Answer

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

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 $\{1 , 2\}$.