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Chapter 4: Problem 87

Solve. $$\left|x^{2}-5\right|=5-x^{2}$$

Short Answer

Expert verified
The solutions are \( x = \pm \sqrt{5} \).

Step by step solution

01

Understand the Absolute Value Equation

The given equation is \ \ \( \left|x^{2}-5\right|=5-x^{2} \). Remember that an absolute value equation \( \left|a\right| = b \) means that either \( a = b \) or \( a = -b \).
02

Set Up Two Separate Equations

Since \( \left|x^{2}-5\right| = 5-x^{2} \), we have two cases: \ \ Case 1: \( x^{2} - 5 = 5 - x^{2} \) \ Case 2: \( x^{2} - 5 = -(5 - x^{2}) \)
03

Solve Case 1

For the first case, solve \( x^{2} - 5 = 5 - x^{2} \): \ \ Add \( x^{2} \) to both sides: \ \( x^{2} + x^{2} - 5 = 5 \) \ \ Combine like terms: \ \( 2x^{2} - 5 = 5 \) \ \ Add 5 to both sides: \ \( 2x^{2} = 10 \) \ \ Divide both sides by 2: \ \( x^{2} = 5 \) \ \ Take the square root of both sides: \ \( x = \pm \sqrt{5} \)
04

Solve Case 2

For the second case, solve \( x^{2} - 5 = -(5 - x^{2}) \): \ \ Distribute the negative sign: \ \( x^{2} - 5 = -5 + x^{2} \) \ \ Subtract \( x^{2} \) from both sides: \ \( x^{2} - x^{2} - 5 = -5 \) \ \ Simplify: \ \( -5 = -5 \). This is true for all real numbers, so it doesn't give us any new solutions.
05

Consolidate the Solutions

The solutions to the original equation come from Case 1. Therefore, the solutions are: \ \( x = \pm \sqrt{5} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root
A square root of a number is a value that, when multiplied by itself, gives the number. The square root symbol is \(\sqrt{}\). There are always two square roots for each positive number: one positive and one negative.

In this exercise, we take the square root of both sides to solve for \(x\):
\(x = \sqrt{5}\ \text{or} \ -\sqrt{5}\)

This gives us two solutions:
\(x = \sqrt{5}\)
\(x = -\sqrt{5}\)
Both satisfy the original absolute value equation.

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