The legs of a right triangle measure 12 and . The hypotenuse measures . What are the lengths of the sides with the unknown values?

### Guidance

In this concept, we will continue solving radical equations. Here, we will address variables and radicals on both sides of the equation.

#### Example A

Solve

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Solution:
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Now we have an
that is not under the radical. We will still isolate the radical.

Now, we can square both sides. Be careful when squaring , the answer is not .

This problem is now a quadratic. To solve quadratic equations, we must either factor, when possible, or use the Quadratic Formula. Combine like terms and set one side equal to zero.

Check both solutions: . 0 is an extraneous solution.

Therefore, 6 is the only solution.

#### Example B

Solve .

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Solution:
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In this example, you need to isolate both radicals. To do this, subtract the second radical from both sides. Then, square both sides to eliminate the variable.

Check:

#### Example C

Solve

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Solution:
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The radical is isolated. To eliminate it, we must raise both sides to the fourth power.

Check:

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Intro Problem Revisit
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Use the Pythagorean Theorem and solve for
*
x
*
then substitute that value in to solve for the sides with unknowns.

Now substitute this value into the sides with the unknowns.

and

Therefore the leg with the unknown measures 5 and the hypotenuse measures 13.

### Guided Practice

Solve the following radical equations. Check for extraneous solutions.

1.

2.

3.

#### Answers

1. The radical is isolated. Cube both sides to eliminate the cubed root.

Check:

2. Square both sides to solve for .

Check:

3. Add to both sides and square to eliminate the radical.

Check both solutions:

5 is an extraneous solution.

### Explore More

Solve the following radical equations. Be sure to check for extraneous solutions.

For questions 13-15, you will need to use the method illustrated in the example below.

- Square both sides
- Combine like terms to isolate the remaining radical
- Square both sides again to solve

Check: Don't forget to check your answers for extraneous solutions!