Transcript 10.6 Radical Equations and Problem Solving 1. Use the power rule to solve radical equations.

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
2
 x 1
2
 5
2
 x
 x  1
x
 x
3a   9a 2
2
3 x 
2
2x  3
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 2  25
Extraneous Solution
x  5,  5
Check all solutions!
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

3y  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
x  5 
2
x5 


x  1
2
x  1 x  1
x 5  x  2 x 1
4 2 x
2
2
2

9 4 1
321
True
4
x
 x
x 4
4 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