Section 7.3

Solving Radical Equations

A radical equation is an equation in which a variable is under a radical. To solve a radical equation:

1. Isolate the radical expression involving the variable. 2. Raise both sides of the equation to the index of the radical. (Remember, if there is no index we use “2”) 3. If there is still a radical equation, repeat Steps 1 and 2; otherwise, solve the resulting equation and check the answer in the original equation .

4. Plug your "x" into the equation to check your work.

Tips & Warnings

* Remember to square both sides! It is the most frequently forgotten step, and the most important. * Make up your own radical equations and try to solve them at home. Practice helps a lot. * Always make sure to check your work. It may help you find any problems in your calculation.

Example #1:

2x−1 =32x−1( )

2= 3( )2

2x−1=9

2x =10x =5

Example #2:

10 −3x=410 −3x( )

2= 4( )2

10 −3x=16

−3x = 6x =−2

Example #3:

x + 3 −1=xx + 3 =x+1

x + 3( )2= x+1( )2

x + 3 =x2 + 2x+1

x + 2 =0 x−1=0

0 =x2 + x−20 = x+ 2( ) x−1( )

x =−2 x=1

−2 + 3 −1 = −2

1 −1 = −2

1−1 = −2

0 ≠ −2

1+ 3 −1=1

4 −1=12 −1=11=1

Check your answers!!

Example #4:

x + 2 −2 =xx + 2 =x+ 2

x + 2( )2= x+ 2( )2

x + 2 =x2 + 4x+ 4

x + 2 =0 x+1=0

0 =x2 + 3x+ 20 = x+ 2( ) x+1( )

x =−2 x=−1

−2 + 2 − 2 = −2

0 − 2 = −2

0 −1 = −2

−2 = −2

−1 + 2 − 2 = −1

1 − 2 = −1

1− 2 = −1

−1 = −1

Check your answers!!

Example #5:

4x−2 − 3x+ 9 =04x−2 = 3x+ 9

4x−2( )2= 3x+ 9( )

2

4x−2 =3x+ 9x−2 =9x =11

Example #6:

2x + 23 =22x + 23( )

3= 2( )3

2x + 2 =8

2x =6

x =3

1) 7x−3 =2

2) 19 −3x−x=−5

Try These…

3) 7x +1 − 6x+ 7 =0

4) 7 −10x3 =3

Try These…