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Q. 79

Expert-verified
Found in: Page 284

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# Solve given equation and verify your result using graphing utility. ${e}^{{x}^{2}}={e}^{3x}.\frac{1}{{e}^{2}}$

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

See the 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.

${x}^{2}=3x-2\phantom{\rule{0ex}{0ex}}{x}^{2}-3x+2=0\phantom{\rule{0ex}{0ex}}$

Factorise above expression.

$\left(x-1\right)\left(x-2\right)=0\phantom{\rule{0ex}{0ex}}\left(x-1\right)=0,\left(x-2\right)=0\phantom{\rule{0ex}{0ex}}x=1,x=2\phantom{\rule{0ex}{0ex}}$

## Step 4. Verification using graphing utility

Graph

${Y}_{1}={e}^{{x}^{2}}\phantom{\rule{0ex}{0ex}}{Y}_{2}={e}^{3x-2}$

Determine the point of intersection.

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

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