Download 6.6 Solve Radical Equations

6.6
Solve Radical Equations
Goal p Solve radical equations.
Your Notes
VOCABULARY
Radical equation
SOLVING RADICAL EQUATIONS
To solve a radical equation, follow these steps:
the radical on one side of the
equation, if necessary.
Step 1
Step 2 Raise each side of the equation to the same
to eliminate the radical and obtain a
linear, quadratic, or other polynomial equation.
the polynomial equation using
techniques you learned in previous chapters.
Check your solution.
Step 3
Example 1
Solve a radical equation
}
Solve Ïx 1 6 5 3.
}
Ïx 1 6 5 3
Write original equation.
5
Square each side to eliminate the
radical.
5
Simplify.
5
Subtract
The solution is
from each side.
. Check this in the original equation.
Checkpoint Solve the equation. Check your solution.
3}
1. Ïx 2 5 1 1 5 21
178
Lesson 6.6 • Algebra 2 Notetaking Guide
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6.6
Solve Radical Equations
Goal p Solve radical equations.
Your Notes
VOCABULARY
Radical equation An equation with a radical that
has variables in the radicand
SOLVING RADICAL EQUATIONS
To solve a radical equation, follow these steps:
Step 1 Isolate the radical on one side of the
equation, if necessary.
Step 2 Raise each side of the equation to the same
power to eliminate the radical and obtain a
linear, quadratic, or other polynomial equation.
Step 3 Solve the polynomial equation using
techniques you learned in previous chapters.
Check your solution.
Example 1
Solve a radical equation
}
Solve Ïx 1 6 5 3.
}
Ïx 1 6 5 3
}
( Ï x 1 6 ) 2 5 32
x16 5 9
x 5 3
Write original equation.
Square each side to eliminate the
radical.
Simplify.
Subtract 6 from each side.
The solution is 3 . Check this in the original equation.
Checkpoint Solve the equation. Check your solution.
3}
1. Ïx 2 5 1 1 5 21
23
178
Lesson 6.6 • Algebra 2 Notetaking Guide
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Your Notes
Solve an equation with a rational exponent
Example 2
(3x 1 4)2/3 5 16
Original equation
Raise each side to the
5
3
power }2 .
5
Apply properties of
exponents.
5
Simplify.
5
Subtract
5
Divide each side by
.
. Check this in the original equation.
The solution is
Solve an equation with an extraneous solution
Example 3
}
x 2 2 5 Ï x 1 10
50
CHECK
Check x 5
Original equation
5
Square each side.
5
Expand left side and
simplify right side.
50
Write in standard
form.
50
Factor.
or
x5
or
.
}
x 2 2 5 Ï x 1 10
0
0
The only solution is
extraneous.)
Copyright © Holt McDougal. All rights reserved.
from each side.
50
x5
Zero product property
Solve for x.
Check x 5 2
.
}
x 2 2 5 Ï x 1 10
0
0
. (The apparent solution
Lesson 6.6 • Algebra 2 Notetaking Guide
is
179
Your Notes
Solve an equation with a rational exponent
Example 2
(3x 1 4)2/3 5 16
Original equation
Raise each side to the
[(3x 1 4)2/3]3/2 5 163/2
3
power }2 .
3x 1 4 5 (161/2)3
Apply properties of
exponents.
3x 1 4 5 64
Simplify.
3x 5 60
Subtract 4 from each side.
x 5 20
Divide each side by 3 .
The solution is 20 . Check this in the original equation.
Solve an equation with an extraneous solution
Example 3
}
x 2 2 5 Ï x 1 10
Original equation
}
(x 2 2)2 5 (Ï x 1 10 )2
Square each side.
x 2 2 4x 1 4 5 x 1 10
Expand left side and
simplify right side.
x 2 2 5x 2 6 5 0
Write in standard
form.
(x 2 6)(x 1 1) 5 0
Factor.
x26 50
Zero product property
or x 1 1 5 0
x 5 6 or
CHECK
Check x 5 6 .
}
x 2 2 5 Ï x 1 10
}
6 2 2 0 Ï6 1 10
}
4 0 Ï16
4 5 4
x 5 21
Solve for x.
Check x 5 2 1 .
}
x 2 2 5 Ï x 1 10
}
21 2 2 0 Ï 21 1 10
}
23 0 Ï 9
23 Þ 3
The only solution is 4 . (The apparent solution 21 is
extraneous.)
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Lesson 6.6 • Algebra 2 Notetaking Guide
179
Your Notes
Example 4
Solve an equation with two radicals
}
}
Solve Ïx 1 6 1 2 5 Ï 10 2 3x .
}
}
Ïx 1 6 1 2 5 Ï10 2 3x
5
5
5
50
x5
CHECK
5
Divide each
side by 4.
5
Square each
side again.
5
Simplify.
05
Write in
standard form.
05
Factor.
50
or
x5
or
Check x 5
Write original
equation.
Square each
side.
Expand left
side and
simplify right
side.
Isolate radical
expression.
Zero product
property
Solve for x.
.
0
0
5
Check x 5 2
.
0
0
5
The only solution is
is extraneous.)
180
Lesson 6.6 • Algebra 2 Notetaking Guide
. (The apparent solution
Copyright © Holt McDougal. All rights reserved.
Your Notes
Solve an equation with two radicals
Example 4
}
}
Solve Ïx 1 6 1 2 5 Ï 10 2 3x .
}
}
Ïx 1 6 1 2 5 Ï10 2 3x
}
}
(Ïx 1 6 1 2)2 5 (Ï10 2 3x )2
}
x 1 6 1 4Ïx 1 6 1 4 5 10 2 3x
}
4Ï x 1 6 5 24x
}
Ïx 1 6 5 2x
}
(Ïx 1 6 )2 5 (2x)2
x 1 6 5 x2
0 5 x2 2 x 2 6
Write original
equation.
Square each
side.
Expand left
side and
simplify right
side.
Isolate radical
expression.
Divide each
side by 4.
Square each
side again.
Simplify.
Write in
standard form.
0 5 (x 2 3)(x 1 2) Factor.
x23 50
or
x 5 3 or
CHECK
x12 50
x 5 22
Zero product
property
Solve for x.
Check x 5 3 .
}
}
Ï3 1 6 1 2 0 Ï10 2 3(3)
}
}
Ï9 1 2 0 Ï1
5 5 1
Check x 5 2 22 .
}
}}
Ï(22)1 6 1 2 0 Ï10 2 3(22)
}
}
Ï4 1 2 0 Ï16
4 5 4
The only solution is 22 . (The apparent solution 3
is extraneous.)
180
Lesson 6.6 • Algebra 2 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Solve the equation. Check for extraneous
solutions.
2. 22x 4/3 2 21 5 253
}
3. x 1 2 5 Ï2x 1 7
}
}
4. Ï 3x 1 4 2 1 5 Ï x 1 5
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 6.6 • Algebra 2 Notetaking Guide
181
Your Notes
Checkpoint Solve the equation. Check for extraneous
solutions.
2. 22x 4/3 2 21 5 253
8
}
3. x 1 2 5 Ï2x 1 7
1
}
}
4. Ï 3x 1 4 2 1 5 Ï x 1 5
4
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 6.6 • Algebra 2 Notetaking Guide
181