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Chapter 8: Problem 6

Solve each equation. $$ \sqrt{x-12}=3 $$

Short Answer

Expert verified
x = 21

Step by step solution

01

Isolate the square root

The equation given is \( \sqrt{x-12}=3 \). The square root is already isolated on one side of the equation.
02

Square both sides

To eliminate the square root, square both sides of the equation: \( (\sqrt{x-12})^2 = 3^2 \). Simplifying this, we get \( x-12 = 9 \).
03

Solve for x

To solve for \( x \), add 12 to both sides of the equation: \( x-12 + 12 = 9 + 12 \). This simplifies to \( x = 21 \).
04

Check the solution

Substitute \( x = 21 \) back into the original equation to verify: \( \sqrt{21-12}=3 \). Simplifying this, we get \( \sqrt{9}=3 \), which is true.

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Key Concepts

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

Isolate the square root
To solve a square root equation, the first crucial step is to isolate the square root expression on one side. This means you need to get the square root term by itself.
In the given equation, \( \sqrt{x-12}=3 \), you can see that the square root is already isolated.
This makes the job easier as we do not need to perform any additional steps to separate it from other terms or constants.
Square both sides
Once the square root is isolated, the next step is to remove it by squaring both sides of the equation. Squaring is necessary because it reverses the square root, allowing us to solve for the variable within.
In our example, we square both sides of the equation:
\( (\sqrt{x-12})^2=3^2 \)

This simplifies to: \( x-12=9 \)
The square root and squared operations cancel each other out on the left side, leaving you with a simple linear equation.
Check the solution
The final step is to always verify your solution. Plugging the value back into the original equation ensures the correctness of your answer.
We found \( x=21 \) in the previous step. Now, substitute it back into the original equation:
\( \sqrt{21-12}=3 \)
Simplify the expression under the square root:
\( \sqrt{9}=3 \)
Since \( 3=3 \), our solution is accurate.
  • Remember, checking the solution helps to identify any possible errors.
  • This step confirms that no extraneous solutions鈥攙alues that do not satisfy the original equation鈥攁re present.

In conclusion, isolating the square root, squaring both sides, and then checking the solution form a reliable method to solve square root equations.

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