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Chapter 3: Problem 18

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$e^{x^{2}-3}=e^{x-2}$$

Short Answer

Expert verified
The solutions are x = 1.618 and x = -0.618 when rounded to three decimal places.

Step by step solution

01

Setup the equation

Since both sides of the equation are exponents of the same base 'e', we can set the exponents equal to each other: \(x^{2} - 3 = x - 2\).
02

Solve for x

Rearrange the equation to a standard quadratic form: \(x^{2} - x - 1 = 0\).
03

Use Quadratic Formula

Solve for 'x' using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Here, a=1, b=-1, and c=-1, so the solutions will be \(x = \frac{1 \pm \sqrt{1+4}}{2}\).
04

Calculation

Calculate the roots, we get two roots as x = 1.618033988749895 and x=-0.618033988749895.
05

Round to three decimal places

Finally, round off the roots to three decimal places. So, the answers are x = 1.618 and x = -0.618.

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