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Chapter 10: Problem 7

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

Short Answer

Expert verified
The solution is \( x = 11 \).

Step by step solution

01

Write the original equation

The given equation is \( \sqrt{x-2}=3 \).
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

Add 2 to both sides to isolate \( x \): \( x - 2 + 2 = 9 + 2 \). This simplifies to \( x = 11 \).
04

Verify the solution

Substitute the value of \( x \) back into the original equation to verify it: \( \sqrt{11-2} = 3 \). Simplifying inside the square root gives \( \sqrt{9} = 3 \), and since \( 3 = 3 \), the solution is correct.

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Key Concepts

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

solving equations
When solving equations, the goal is to find the value of the variable that makes the equation true. Here, we are given an equation that involves a square root.
The equation \(\textstyle{\boxed{\textstyle\textstyle{\sqrt{x-2}=3}}}\) needs to be solved to find the value of \(x\). The basic steps for solving almost any equation are:
  • Isolate the variable.
  • Simplify each side of the equation as much as possible.
  • Use inverse operations to solve for the variable.

In this problem, isolating the square root term \(\textstyle\sqrt{x-2}\) was the first step. By squaring both sides, we removed the square root and simplified the equation to solve for \(x\).
square roots
Square roots are the opposite of squaring a number. The square root of a number \(n\) is a value that, when multiplied by itself, gives \(n\). For example, the square root of 9 is 3, because \(3 \times 3 = 9\).
The square root symbol \(\sqrt{}\) is used to represent the operation.
Key things to remember:
  • Squaring removes the square root - \( (\sqrt{a})^2 = a \).
  • Always consider both the positive and negative roots - \( \sqrt{9} = \pm 3 \).

For our example problem, \( \sqrt{x-2} = 3 \), squaring both sides simplifies it to \( x-2 = 9 \), making it easier to isolate \(x\) and find its value.
verification of solutions
Verification of solutions is a crucial step in solving equations. It ensures that the solution found is correct and satisfies the original equation.
To verify our solution for \( \sqrt{x-2} = 3 \), we substitute \( x = 11 \) back into the original equation:
\( \sqrt{11-2} = 3 \).
Simplifying inside the square root, we get \( \sqrt{9} = 3 \). Since \( 3 = 3 \), our solution is confirmed to be correct.
Always follow these steps for verification:
  • Substitute the found solution back into the original equation.
  • Simplify both sides of the equation.
  • Check if both sides of the equation are equal.

This practice helps avoid mistakes and confirms that the process followed was accurate.

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