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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 108 (page 1019)

In the following exercises, solve each equation.
$\frac{e^{x^{2}}}{e^{3}} = e^{2 x}$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given

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

02

Step 2. Solve the equation

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

Bases are same so the exponents should be equal.

$x^{2} - 3 = 2 x x^{2} - 2 x - 3 = 0 x^{2} - 3 x + x - 3 = 0 x (x - 3) + 1 (x - 3) = 0 (x - 3) (x + 1) = 0$

So,

$x = 3 x = - 1$

03

Step 3. Check the answers

Put $x = 3$ in given equation,

$\frac{e^{x^{2}}}{e^{3}} = e^{2 x} \frac{e^{{(3)}^{2}}}{e^{3}} = e^{2 (3)} \frac{e^{9}}{e^{3}} = e^{6} e^{9 - 3} = e^{6} e^{6} = e^{6}$

True

Put $x = - 1$ in given equation,

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

True

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