chapter 8 section 6 copyright © 2008 pearson education, inc. publishing as pearson addison-wesley
TRANSCRIPT
Chapter Chapter 88Section Section 66
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.
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8.68.68.68.6
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
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Objective 11
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.
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EXAMPLE 1
Solve.
Solution:
Using the Squaring Property of Equality
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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
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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
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Objective 22
Identify equations with no solutions.
Slide 8.6 - 8
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EXAMPLE 3
Solution:
Using the Squaring Property when One Side Is Negative
Slide 8.6 - 9
Solve.4x
2 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:
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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.
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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.
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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.
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Objective 33
Slide 8.6 - 12
Solve equations by squaring a binomial.
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EXAMPLE 5
Solve
Solution:
Using the Squaring Property when One Side Has Two Terms
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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 0x 2 1 0x 4x 1
2x
Since x must be positive the solution set is {4}.
or
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Solve.
EXAMPLE 6 Rewriting an Equation before using the Squaring Property
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Solution:
25 6x x
625 66x x
2 2
25 6x x 225 12 325 256x x xx x 20 13 36x x
0 4 9x x 0 9x 0 4x
9x 4x
The solution set is {4,9}.
or
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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.
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9 2 1x x
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EXAMPLE 7 Using the Squaring Property Twice
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Solve.
Solution:
1 4 1x x
1 1 4x x
2 2
1 1 4x x
1 1 2 4 4x x x
224 2 4x
16 4 16x 32
4 4
4x
8x The solution set is {8}.
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Objective 44
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.
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EXAMPLE 8Solving Equations with Cube Root Radicals
Slide 8.6 - 19
Solve each equation.
Solution:
3 37 4 2x x 3 2 3 26 27x x
3 3
3 2 3 26 27x x 2 26 27x x
20 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
3
27,1
or