1. 7.7 Solving Radical Equations
p.453

2. What is a Radical Expression?
• A Radical Expression
is an equation that
has a variable in a
radicand or has a
variable with a
rational exponent.
3
x 10
2
3
yes
25
yes
3 x 10
no
( x 2)

3. Steps to solve a radical equation:
STEP 1: Isolate the radical on one side of the equation
(if possible)
STEP 2: Raise each side of the equation to a power equal to the
index of the radical to eliminate the radical
STEP 3: Solve the remaining polynomial equation.
CHECK YOUR RESULTS.

4. EXAMPLE – Solving a Radical
Equation
5x 1 6 0
5x 1 6
2
2
5x 1 (6 )
5x 1 36
5x 35
x 7
Square both sides to get rid of
the square root

5. EXAMPLE
x 15
3
x
2
2
x 15 (3
x 15 (3
x 15
24
3
16
x)
x )(3
9 6 x
15
16 15
x)
x
NO SOLUTION
Since 16 doesn’t plug in
as a solution.
9 6 x
6 x
4
x
16 x
1 3 4
1 1
Let’s Double Check
that this works
Note: You will get
Extraneous
Solutions from time
to time – always do
a quick check

6. Let’s Try Some
2
3x 2
6
2( x 2)
2
3
50

7. Let’s Try Some
2
3x 2
6
2( x 2)
2
3
50

8. SOLVING MORE COMPLEX
EQUATIONS
4( x 1)
2
101
20
2
4( x 1)
2
( x 1)
( x 1)
+ because we
are taking an
even power
(square root)
of both sides
81
81
( x 1) i
( x 1)
( x 1)
9i
x 1
9i
2
2
4
81
4
81
1
4
i 92
9i
2
1
9i
2

9. RADICAL EQUATIONS
3(5n 1)
1
3
2
3(5n 1)
1
3
2
1
3
(5n 1)
1 3
(5n 1) 3
(5n 1)
5n
5n
8
0
2
8
27
5n
RAISE BOTH SIDES
TO RECIPROCAL POWER
3
2
27
8
ISOLATE RADICAL / RATIONAL
3
3
SOLVE FOR THE VARIABLE
27
1
27
35
27
27
n
7
27

10. SOLVING MORE COMPLEX
EQUATIONS
(2 x 1)
0.5
(3x 4)
0.25
0
0.5
0.25
(2 x 1)
(3x 4)
0.5 4
0.25 4
[( 2 x 1) ] [(3x 4) ]
2
(2 x 1) 3 x 4
4x
2
4 x 1 3x 4
4x
2
Raise each side to the 4th
power. This will get you
integer powers – much
easier to work with!
x
x 3 0
3
, 1
4
Factor

11. Let’s Try Some . . . check for
extraneous solutions
(2 x 1)
0.5
(3x 4)
0.25
0

12. Let’s Try Some . . . check for
extraneous solutions
(2 x 1)
0.5
(3x 4)
0.25
0
x
3
, 1
4

13. Can graphing calculators help?
SURE!
x
1.
2.
3.
4.
x 2
Input x for Y1
Input x-2 for Y2
Graph
Find the points of
intersection
One Solution at (4, 2)
To see if this is extraneous or not, plug the x value
back into the equation. Does it work?