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10.6
Radical Equations and Problem Solving
1. Use the power rule to solve radical equations.
 7

7 7
49  7

5 5
25  5
2

x x
2
 x 1
2
 5
2
 x
 x  1
3a 
2
3 x 
2
2x  3
2
 x
 9a 2
 9x
 4 x 2  12 x  9
Not 4x2 + 9 !
To Solve Equations: Inverse Operations
Add ↔ Subtract
Multiply ↔ Divide
Powers↔ Roots
Radical Equations
x 
x 
Power Rule
If both sides of an equation are raised to the same
power, all solutions of the original equation are also
solutions of the new equation.
x  5
x  25
Extraneous Solution
x  5,  5
Check all solutions!
2
Radical Equations
x 5
y 6
x  25
y  36
3
a  2
a  8
4
b 2
b  16
To Solve Radical Equations
1. Isolate the radical.
2. Raise both sides of the equation to the same
power as the index.
3. Linear or Quadratic? Solve. (Repeat 1 & 2 if
necessary)
4. Check.
Solve:
3y  2  3  2
3y  2  5

3 y  2   5 
2
Check:
39  2  3  2
2
25  3  2
3 y  2  25
3 y  27
y 9
True
9 
Solve:
3

3
k2 4
k  2   4
3
k  2  64
3
Check:
3
62  2  4
3
64  4
k  62
True
62
Solve:
c371
c  3  6
Square root will never be negative!
Solve:
m2 

m  2 
2
2m  3

2m  3 
m  2  2m  3
 m  5
m 5
Check:
2
52 
7 
True
5
25  3
7
Solve:

7  3x  x  3
Rewrite and Foil!
7  3 x   x  3 
2
Check: x = -1
2
7  3 1   1  3
7  3x  x 2  6x  9
4 2
0  x  3x  2
True
2
0  x  1x  2
x  1 x  2
 1,
 2
Check: x = -2
7  3 2   2  3
11
True
Solve:

Check: y = 2
4y  1  5  y
42  1  5  2
4y  1  y  5
952
4y  1  y  5
2
2
4y  1  y  10y  25
2
0  y  14y  24
2
0  y  2y  12
y 2
Extraneous Solution
y  12
 12 
35 2
False
Check: y = 12
412  1  5  12
49  5  12
7  5  12
True
Solve:

x5
x 1
Check:
x5 
x 1
45 4 1
x  5 
2
x5 


x  1
2
x  1 x  1
x 5  x  2 x 1
42 x
2
2
2

True
4
x
 x
x 4
9 4 1
321
2
Solve.
5x  4  x  2
a) 6
b) 8
c) 9
d) no solution
Copyright © 2011 Pearson Education, Inc.
Slide 10- 13
Solve.
5x  4  x  2
a) 6
b) 8
c) 9
d) no solution
Copyright © 2011 Pearson Education, Inc.
Slide 10- 14
Solve. x  2  5x  16
a) 3, 4
b) 3
c) 4
d) no real-number solution
Copyright © 2011 Pearson Education, Inc.
Slide 10- 15
Solve. x  2  5x  16
a) 3, 4
b) 3
c) 4
d) no real-number solution
Copyright © 2011 Pearson Education, Inc.
Slide 10- 16