2. Roots, Radicals, and Root Functions
Chapter 9
Roots, Radicals, and Root Functions

3. Solving Equations with Radicals
9.6
Solving Equations with Radicals

4. 9.6 Solving Equations with Radicals
Objectives
Solve radical equations using the power rule.
Solve radical equations that require additional steps.
Solve radical equations with indexes greater than 2.
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5. 9.6 Solving Equations with Radicals
Power Rule for Solving Equations with Radicals
Power Rule for Solving Equations with Radicals
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.
Read the power rule carefully; it does not say that all solutions of the new
equation are solutions of the original equation. They may or may not be.
Solutions that do not satisfy the original equation are called extraneous
solutions; they must be discarded.
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6. 9.6 Solving Equations with Radicals
Caution on Checking Solutions
CAUTION
When the power rule is used to solve an equation, every solution of the
new equation must be checked in the original equation.
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7. 9.6 Solving Equations with Radicals
EXAMPLE 1
Using the Power Rule
Solve
5x – = 7.
Use the power rule and square both sides to get
5x – = 7 2
5x – = 49
5x = 55
Add 6.
x = 11
Divide by 5.
To check, substitute the potential solution in the original equation.
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8. 9.6 Solving Equations with Radicals
EXAMPLE 1
Using the Power Rule
Solve
5x – = 7.
5x – = 7
5 · 11 – = ? Let x = 11.
7 = True
Since 11 satisfies the original equation, the solution set is { 11 }.
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9. 9.6 Solving Equations with Radicals
Solving an Equation with Radicals
Solving an Equation with Radicals
Step Isolate the radical. Make sure that one radical term is alone on
one side of the equation.
Step Apply the power rule. Raise both sides of the equation to a
power that is the same as the index of the radical.
Step Solve. Solve the resulting equation; if it still contains a radical,
repeat Steps 1 and 2.
Step Check all potential solutions in the original equation.
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10. 9.6 Solving Equations with Radicals
Caution on Incorrect Solution Sets
CAUTION
Remember Step 4 or you may get an incorrect solution set.
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11. 9.6 Solving Equations with Radicals
EXAMPLE 2
Using the Power Rule
Solve
2x – = 0.
Step To isolate the radical on one side, subtract 1 from each side.
2x – = –1
Step Now square both sides.
2x – = (–1) 2
Step Solve.
2x – = 1
2x = 10
x = 5
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12. 9.6 Solving Equations with Radicals
EXAMPLE 2
Using the Power Rule
Solve
2x – = 0.
Step Check the potential solution, 5, by substituting it in the original
equation.
2x – = 0
2 · 5 – = ? Let x = 5.
= False
This false result shows that 5 is not a solution of the original equation; it is
extraneous. The solution set is ∅.
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13. 9.6 Solving Equations with Radicals
Note on Identifying No Solution
NOTE
We could have determined after Step 1 that the equation in Example 2
has no solution because the expression on the left cannot be negative.
EXAMPLE 2
2x – = 0.
2x – = –1
Step To isolate the radical on one side, subtract 1 from each side.
Solve
The expression on the left cannot be negative.
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14. 9.6 Solving Equations with Radicals
Finding the Square of a Binomial
Finding the Square of a Binomial
Recall that
(x + y)2 = x xy + y2.
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15. 9.6 Solving Equations with Radicals
EXAMPLE 3
Using the Power Rule; Squaring a Binomial
Solve
9 – x = x
Step The radical is alone on the left side of the equation.
Step Now square both sides.
9 – x = (x + 3) 2
9 – x = x x + 9
Step The new equation is quadratic, so get 0 on one side.
Twice the product of 3 and x.
0 = x x Subtract 9 and add x.
0 = x(x + 7) Factor.
x = or x = Zero-factor property
x = –7
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16. 9.6 Solving Equations with Radicals
EXAMPLE 3
Using the Power Rule; Squaring a Binomial
Solve
9 – x = x
Step Check each potential solution, 0 and –7, in the original equation.
If x = 0, then
If x = –7, then
9 – x = x + 3
9 – x = x + 3
9 – = ?
9 – (–7) = (–7) ?
= 3 ?
= –4 ?
= 3 True
= – False
The solution set is { 0 }. The other potential solution, –7, is extraneous.
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17. 9.6 Solving Equations with Radicals
Caution on the Middle Term
CAUTION
When a radical equation requires squaring a binomial as in Example 3,
remember to include the middle term.
(x + 3)2 = x x + 9
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18. 9.6 Solving Equations with Radicals
EXAMPLE 4
Using the Power Rule; Squaring a Binomial
Solve
x2 – 8x = x – 5.
Square both sides. 2
x2 – 8x = (x – 5)
x2 – 8x = x2 – 10x
Twice the product of 5 and x.
2x = Subtract x2 and 3; add 10x.
x = Divide by 2.
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19. 9.6 Solving Equations with Radicals
EXAMPLE 4
Using the Power Rule; Squaring a Binomial
Solve
x2 – 8x = x – 5.
Check:
x2 – 8x = x – 5
112 – 8 · = – ? Let x = 11.
6 = True
The solution set of the original equation is { 11 }.
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20. 9.6 Solving Equations with Radicals
EXAMPLE 5
Using the Power Rule; Squaring Twice
Solve
x –x = 5.
Start by isolating one radical on one side of the equation by subtracting
from each side. Then square both sides.
–x + 6
x –x = 5
x + 7
= 5 – –x + 6 2
x + 7
= – –x + 6
= – –x (–x + 6)
x + 7
Twice the product of 5 and
–x + 6
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21. 9.6 Solving Equations with Radicals
EXAMPLE 5
Using the Power Rule; Squaring Twice
Solve
x –x = 5.
This equation still contains a radical, so square both sides again. Before doing
this, isolate the radical term on the right.
= – –x (–x + 6)
x + 7
= – x – –x + 6
x + 7
= – –x + 6
2x – 24
Subtract 31 and add x.
= – –x + 6
x – 12
Divide by 2.
= – –x + 6
(x – 12)2
Square both sides again. 2
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22. 9.6 Solving Equations with Radicals
EXAMPLE 5
Using the Power Rule; Squaring Twice
Solve
x –x = 5.
This equation still contains a radical, so square both sides again.
= – –x + 6
(x – 12)2
Square both sides again. 2
= (–5) –x + 6
x2 – 24x
(ab)2 = a2 b2 2
= ( –x )
x2 – 24x
= –25x
x2 – 24x
Distributive property =
x2 + x – 6
Standard form =
(x + 3)(x – 2)
Factor.
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23. 9.6 Solving Equations with Radicals
EXAMPLE 5
Using the Power Rule; Squaring Twice
Solve
x –x = 5.
Now finish solving the equation. =
(x + 3)(x – 2)
x = or x – 2 = 0
Zero-factor property
x = –3 or x = 2
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24. 9.6 Solving Equations with Radicals
EXAMPLE 5
Using the Power Rule; Squaring Twice
Solve
x –x = 5.
Check each potential solution, –3 and 2, in the original equation.
If x = –3, then
If x = 2, then
x –x = 5
x –x = 5
– –(–3) = 5
–(2) = 5
= 5
= 5
5 = 5
5 = 5
The solution set is { –3, 2 }.
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25. 9.6 Solving Equations with Radicals
EXAMPLE 6
Using the Power Rule for a Power Greater
than 2
3x – = x 3
Solve
Raise both sides to the third power.
3x – = x + 6 3
3x – 2 = 2x + 6
x = 8
Check this result in the original equation.
3x – = x + 6 3
3 · 8 – = · ? Let x = 8. 3
22 = True 3
The solution set is { 8 }.
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