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Chapter 8: Problem 3

Solve each equation. $$ \sqrt{x+2}=3 $$

Short Answer

Expert verified
The solution is \( x = 7 \).

Step by step solution

01

Isolate the square root

The given equation is \( \sqrt{x + 2} = 3 \). Since the square root is already isolated, can move to the next step.
02

Square both sides

To eliminate the square root, square both sides of the equation. \[ (\sqrt{x + 2})^2 = 3^2 \] This simplifies to: \[ x + 2 = 9 \]
03

Solve for \(x\)

Now, isolate \( x \) by subtracting 2 from both sides of the equation. \[ x + 2 - 2 = 9 - 2 \] Simplifying this gives: \[ x = 7 \]
04

Verify the solution

Substitute \( x = 7 \) back into the original equation to verify: \( \sqrt{7 + 2} = 3 \) This simplifies to: \( \sqrt{9} = 3 \) Since \( 3 = 3 \), the solution is verified.

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Key Concepts

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

Isolating Variables
When solving square root equations, the first step is to isolate the square root on one side of the equation. This process is called 'isolating the variable'. In many cases, you might need to move other terms to the opposite side of the equation to achieve this.
However, in the given exercise, the square root is already isolated: \(\, \sqrt{x + 2} = 3 \) Since there are no additional terms on the same side as the square root, you can proceed directly to the next step.
Here are some simple steps to isolate a square root if needed:
  • Identify the term with the square root.
  • Move any added or subtracted terms to the other side of the equation.
  • Divide or multiply both sides to leave the square root term on its own.
Squaring Both Sides
The next step in solving square root equations is to eliminate the square root by squaring both sides of the equation. This helps to simplify the equation and makes it easier to solve for the variable.
For our example, starting from \(\sqrt{x + 2} = 3\), we square both sides: \[ (\sqrt{x + 2})^2 = 3^2 \]
This results in:\

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