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Chapter 1: Problem 39

Find the real solutions, if any, of each equation. $$ \left(x^{2}+9\right)^{1 / 2}=5 $$

Short Answer

Expert verified
The real solutions are \( x = 4 \) and \( x = -4 \).

Step by step solution

01

Isolate the square root

Given equation is \( \left(x^{2}+9\right)^{1 / 2}=5 \). To isolate the square root, take the square of both sides: \[ \left(\left(x^{2}+9\right)^{1 / 2}\right)^2 = 5^2 \].
02

Simplify both sides

Simplify the equation from Step 1. The left side becomes \( x^2 + 9 \) and the right side becomes 25, resulting in the equation \( x^2 + 9 = 25 \).
03

Solve for \( x^2 \)

Subtract 9 from both sides to isolate \( x^2 \): \[ x^2 = 25 - 9 \]. This simplifies to \( x^2 = 16 \).
04

Solve for \( x \)

Take the square root of both sides: \[ x = \pm \sqrt{16} \]. This results in \( x = 4 \) or \( x = -4 \).

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Key Concepts

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

Square Root Isolation
To solve any equation that involves a square root, the first step is to isolate the square root term. In the equation \( (x^{2}+9)^{1 / 2}=5 \), the square root term \( (x^{2}+9)^{1 / 2} \) is already isolated on one side.
Next, we need to eliminate the square root by squaring both sides of the equation. This step is crucial because it helps us transform our original equation into one without square roots.
  • Square both sides: \[ \big((x^2 + 9)^{1/2}\big)^2 = 5^2 \]
  • The left side simplifies to \ x^2 + 9 \ and the right side simplifies to \ 25 \.
Now our equation looks like \ x^2 + 9 = 25 \.
Simplifying Equations
Once the square root has been removed, we are left with a simpler quadratic equation. The next step is to simplify this equation to make it easier to solve.
In this exercise, we have \ x^2 + 9 = 25 \ after isolating the square root and squaring both sides.
  • First, subtract 9 from both sides to isolate the \ x^2 \ term.
  • This gives us: \[ x^2 = 25 - 9 \]
  • Simplify the right side: \[ x^2 = 16 \]
Now we have a basic quadratic equation \ x^2 = 16 \ that is ready to be solved.
Solving for x
The final step in solving the quadratic equation \ x^2 = 16 \ is to find the values of \ x \. This involves taking the square root of both sides.
  • Take the square root of both sides: \[ x = \pm \sqrt{16} \]
  • This results in two possible solutions: \ x = 4 \ and \ x = -4 \.
It's important to recognize that squaring both sides can sometimes introduce extraneous solutions, but in this case, both solutions are valid.
So the real solutions to the equation \ (x^2 + 9)^{1/2} = 5 \ are \ x = 4 \ and \ x = -4 \.

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