Chapter 8
Section 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Equations with Radicals
Solve radical equations having square root
radicals.
Identify equations with no solutions.
Solve equations by squaring a binomial.
Solve radical equations having cube root
radicals.
1
4
3
2
8.6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Equations with Radicals.
A radical equation is an equation having a variable
in the radicand, such as
Slide 8.6 - 3
1 3x or 3 8 9x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Slide 8.6 - 4
Solve radical equations having
square root radicals.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To solve radical equations having square root
radicals, we need a new property, called the squaring
property of equality.
Be very careful with the squaring property: Using this property can give a
new equation with more solutions than the original equation has. Because of this
possibility, checking is an essential part of the process. All proposed solutions
from the squared equation must be checked in the original equation.
Slide 8.6 - 5
Solve radical equations having
square root radicals.
If each side of a given equation is squared, then all
solutions of the original equation are among the
solutions of the squared equation.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Solve.
Solution:
Using the Squaring
Property of Equality
Slide 8.6 - 6
It is important to note that even though the algebraic work may be done
perfectly, the answer produced may not make the original equation true.
9 4x
229 4x
9 16x
9 169 9x
7x
7x 7
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve.
EXAMPLE 2 Using the Squaring Property
with a Radical on Each Side
Slide 8.6 - 7
Solution:
3 9 2x x
2 2
3 9 2x x
3 9 4x x
3 33 9 4xx x x
9x
9
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 2
Identify equations with no
solutions.
Slide 8.6 - 8
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
Using the Squaring Property
when One Side Is Negative
Slide 8.6 - 9
Solve.
4x2 2
4x
16x
16 4
4 4
4x
False
Because represents the principal or nonnegative square root of x in Example 3,
we might have seen immediately that there is no solution.
x
Check:
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use the following steps when solving an equation with
radicals.
Step 1 Isolate a radical. Arrange the terms so that
a radical is isolated on one side of the
equation.
Solving a Radical Equation.
Slide 8.6 - 10
Step 6 Check all proposed solutions in the original
equation.
Step 5 Solve the equation. Find all proposed solutions.
Step 4 Repeat Steps 1-3 if there is still a term with a
radical.
Step 3 Combine like terms.
Step 2 Square both sides.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solution:
Using the Squaring Property
with a Quadratic Expression
Slide 8.6 - 11
Solve 2 4 16.x x x
22 2 4 16x x x
2 22 24 16x xx x x
44 40 16xx x
4 1
4 4
6x
4x
Since x must be a positive number the
solution set is Ø.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 3
Slide 8.6 - 12
Solve equations by squaring a
binomial.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Solve
Solution:
Using the Squaring Property
when One Side Has Two Terms
Slide 8.6 - 13
2 1 10 9.x x
222 1 10 9x x
2 10 94 4 1 10 99 10x x xx x
24 14 8 0x x
2 1 2 8 0x x
2 8 0x2 1 0x
4x1
2x
Since x must be positive the solution set is {4}.
or
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve.
EXAMPLE 6 Rewriting an Equation before
using the Squaring Property
Slide 8.6 - 14
Solution:
25 6x x
625 66x x2 2
25 6x x
225 12 325 256x x xx x20 13 36x x
0 4 9x x
0 9x0 4x
9x4x
The solution set is {4,9}.
or
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve equations by squaring a binomial.
Errors often occur when both sides of an equation are squared. For
instance, when both sides of
are squared, the entire binomial 2x + 1 must be squared to get 4x2 + 4x + 1.
It is incorrect to square the 2x and the 1 separately to get 4x2 + 1.
Slide 8.6 - 15
9 2 1x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7
Using the Squaring
Property Twice
Slide 8.6 - 16
Solve.
Solution:
1 4 1x x
1 1 4x x2 2
1 1 4x x
1 1 2 4 4x x x2
24 2 4x
16 4 16x
32
4 4
4x
8xThe solution set is {8}.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 4
Slide 8.6 - 17
Solve radical equations having
cube root radicals.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve radical equations having cube
root radicals.
Slide 8.6 - 18
We can extend the concept of raising both sides of
an equation to a power in order to solve radical
equations with cube roots.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 8 Solving Equations with
Cube Root Radicals
Slide 8.6 - 19
Solve each equation.
Solution:
3 37 4 2x x 3 2 3 26 27x x3 3
3 2 3 26 27x x
2 26 27x x20 26 27x
0 27 1x x
0 27x 0 1x
27x 1x
3 33 37 4 2x x
7 4 2x x
3 2
3 3
x
2
3x
2
327,1
or
Top Related