The legs of a right triangle measure 3 and

### Solving Radical Equations

Solving radical equations are very similar to solving other types of equations. The objective is to get

Let's determine if

Plug in 5 for

We know that

Now, let's solve the following equations for x.

2x−5−−−−−√+7=16

To solve for

Now, we can square both sides to eliminate the radical. Only square both sides when the radical is alone on one side of the equals sign.

Check:

ALWAYS check your answers when solving radical equations. Sometimes, you will solve an equation, get a solution, and then plug it back in and it will not work. These types of solutions are called **extraneous solutions** and are not actually considered solutions to the equation.

3x−8−−−−−√3−2=−14

Again, isolate the radical first. Add 2 to both sides and divide by 3.

Now, cube both sides to eliminate the radical.

Check:

### Examples

#### Example 1

Earlier, you were asked to find the length of the leg with the unknown value.

Use the Pythagorean Theorem and solve for *x* then substitute that value in to solve for the leg with the unknown.

Now substitute this value into the leg with the unknown.

Therefore the leg with the unknown has a length of 4.

#### Example 2

Solve for x:

The radical is already isolated here. Square both sides and solve for

Check:

#### Example 3

Solve for x:

Isolate the radical by subtracting 1 and then dividing by 5.

Square both sides and continue to solve for

Check:

#### Example 4

Solve for x:

In this problem, we have a fourth root. That means, once we isolate the radical, we must raise both sides to the fourth power to eliminate it.

Check:

### Review

Determine if the given values of *x* are solutions to the radical equations below.

x−3−−−−−√=7;x=32 6+x−−−−−√3=3;x=21 2x+3−−−−−√4−11=−9;x=6

Solve the equations and check your answers.

x+5−−−−−√=6 2−x+1−−−−−√=0 45−x−−−−−√=12 x+9−−−−−√+7=11 12x−2−−−−−√3=1 x+3−−−−−√3+5=9 515−x−−−−−√+2=17 −5=x−5−−−−−√5−7 x−6−−−−−√4+10=13 85x+5−−−−−√3=8 3x+7−−−−−√−2=25 235+x−−−−−−√4+9=14

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 7.7.