Download Notes: "Solving Radical Equations"

Objectives:
1. Be able to solve a radical equation.
2. Be able to solve an equation that contains a rational
exponent.
Critical Vocabulary:
Rational Exponents, Extraneous Solutions
I. Solving Radical Equations
Example 1:

5 x  91/ 2  11
5 x  9  11

5 x  9  11
5x  9   11
5x  9  121
5x  9  121
5x  130
5x  130
x  26
x  26
2
2
1/ 2 2
2
I. Solving Radical Equations
Example 2:
2 x  31/ 2  5  8
2 x  31/ 2  13
2x  3  5  8
2 x  3  13

2 x  3  13
2x  3   13
2x  3  169
2x  3  169
2x  166
2x  166
x  83
x  83

2
2
1/ 2 2
2
I. Solving Radical Equations
Example 3:
3

3
2 x  71/ 3  3
2x  7  3
2 x  7  3
2x  7   3
2x  7  27
2x  7  27
2x  20
2x  20
x  10
x  10

3
3
1/ 3 3
3
I. Solving Radical Equations
Example 4:
2 x  11/ 4  8  5
2x  1  8  5
4
4

4
2 x  11/ 4  3
2 x  1  3

2 x  1   3
4
4
2x  1    3
1/ 4 4
2x  1  81
2x  1  81
2x  80
2x  80
x  40
x  40
4
This can never happen, therefore the answer is
NO SOLUTION and you found an extraneous
solution.
II. Solving Rational Exponent Equations
3
4
Example 5: ( x  2)  1  7
4
3
4
3/ 4 4 / 3
x  23  8
4
 4 x  23   84
4
( x  2)  8
( x  2) 
x  23  1  7
 8
4/3


x  23  4096
x  2  3 4096
x  2  4096
3
x  2  16
x  14
3
x  23  3 4096
x  2  3 4096
x  2  3 4096
x  2  16
x  14
II. Solving Rational Exponent Equations
2 1/ 5
x  2
3
 3  2 1/ 5 
 3
    x   2  
 2  3

 2
Example 6: 

25
x  2
3
 3  2 5 
 3


x


2



 
 2  3

 2
x1 / 5  3
x   3
x   3
1/ 5 5
1/ 5 5
5
5
x  243
5
x 3
 x   3
5
5
x  243
5
Page 456 #3-21 odds, 23-31
(15 problems)
Objectives:
1. Be able to solve a radical equation.
2. Be able to solve an equation that contains a rational
exponent.
3. Be able to solve an equation that may contain
extraneous solutions.
4. Be able to solve an equation that contains multiple
radicals.
Critical Vocabulary:
Rational Exponents, Extraneous Solutions
Warm Up:
3
2
4x  8  0
x  2  7  10
WARM UP #1
3
2
4x  8  0
3
2
4x  8
3
2
x 2
2
3
2
 32 
 x   23
 
 
x3 4
Get the variable and its
exponent alone
Raise each side to the 2/3
power
Simplify each side
WARM UP #2
x  2  7  10
x  2  17
x  2  289
x  287
Isolate the root
Get rid of the root
Solve for “x”
III. Solving Equations with Extraneous Solutions
2x  7  x  2
2x  7  x  2
2
2x  7  x  4x  4
0  x2  2x  3
2
0  x  2x  3
0  ( x  3)( x  1)
x  3 x  1
Isolate the root
Get rid of the root
Solve for “x”
Check for extraneous
solutions
III. Solving Equations with Extraneous Solutions
x  4  2x
2
x  8 x  16  2 x
x 2  10 x  16  0
x 2  10 x  16  0
( x  8)( x  2)  0
x 8 x 2
Get rid of the root
Solve for “x”
Check for extraneous
solutions
IV. Solving Equations with Multiple Radicals
2x  5  x  3  1
2x  5  x  3 1
2x  5  x  2 x  3  2
x 3
 x 3
2
x2  6x  9
 x 3
4
x 2  6 x  9  4 x  12
x 2  10 x  21  0
( x  7)( x  3)  0
x7 x 3
Move the root
Square Both Sides
Isolate the Root
Square Both Sides
Solve for “x”
Check for Extraneous
Solutions
Page 457 #34-43, 45-52