10.6 solving equations with radicals. solve radical equations by using the power rule. objective 1...
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10.6 Solving Equations with Radicals
Solve radical equations by using the power rule.
Objective 1
Slide 10.6- 2
Power Rule for Solving an Equation 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.
Slide 10.6- 3
Solve radical equations by using the power rule.
The power rule 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 rejected.
When the power rule is used to solve an equation, every solution of the new equation must be checked in the original equation.
Solve 5 1 4.x
225 1 4x
5 1 16x
5 15x
3x
Check:
5 1 4x
5 3 1 4
16 4
4 4
Since 3 satisfies the original equation, the solution set is {3}.
Slide 10.6- 4
CLASSROOM EXAMPLE 1 Using the Power Rule
Solution:
True
Solving an Equation with Radicals
Step 1 Isolate the radical. Make sure that one radical term is alone on one side of the equation.
Step 2 Apply the power rule. Raise each side of the equation to a power that is the same as the index of the radical.
Step 3 Solve the resulting equation; if it still contains a radical, repeat Steps 1 and 2.
Step 4 Check all proposed solutions in the original equation.
Slide 10.6- 5
Solve radical equations by using the power rule.
Solve 5 3 2 0.x
5 3 2x
The equation has no solution, because the square root of a real number must be nonnegative.The solution set is .
Slide 10.6- 6
CLASSROOM EXAMPLE 2 Using the Power Rule
Solution:
Solve radical equations that require additional steps.
Objective 2
Slide 10.6- 7
Solve 5 1.x x
Step 1 The radical is alone on the left side of the equation.
2 2
5 1x x Step 2 Square both sides.
25 2 1x x x Step 3 The new equation is quadratic, so get 0 on one side.
20 3 4x x
0 ( 4)( 1)x x
4 0 or 1 0x x 4 or 1x x
Slide 10.6- 8
CLASSROOM EXAMPLE 3 Using the Power Rule (Squaring a Binomial)
Solution:
Step 4 Check each proposed solution in the original equation.
4 or 1x x
5 1x x
4( ) 145 ) (
9 3
False
5 1x x
5 (1) (1) 1
4 2
2 23 3True
The solution set is {1}. The other proposed solution, – 4, is extraneous.
Slide 10.6- 9
CLASSROOM EXAMPLE 3 Using the Power Rule (Squaring a Binomial) (cont’d)
Solve 21 2 1.x x x
Step 1 The radical is alone on the left side of the equation.
2
221 2 1x x x Step 2 Square both sides.
2 21 2 2 1x x x x Step 3 The new equation is quadratic, so get 0 on one side.
20 2 4x x 0 2 ( 2)x x
2 0 or 2 0x x 0 or 2x x
Slide 10.6- 10
CLASSROOM EXAMPLE 4 Using the Power Rule (Squaring a Binomial)
Solution:
Step 4 Check each proposed solution in the original equation.
0 or 2x x
21 2 1x x x
21 2(0) 00 1
1 1
False
21 2 1x x x
21 2( 2) ( 2) 2 1
1 11 1True
The solution set of the original equation is {0}.
Slide 10.6- 11
CLASSROOM EXAMPLE 4 Using the Power Rule (Squaring a Binomial) (cont’d)
Solve 2 3 1 1.x x
2 3 1 1x x
2 2
2 3 1 1x x Square both sides.
2 3 1 2 1 ( 1)x x x
1 2 1x x Isolate the remaining radical.
221 2 1x x Square both sides.
22 22 1 ( 2) 1x x x 2 2 1 4( 1)x x x
Apply the exponent rule (ab)2 = a2b2.
Slide 10.6- 12
CLASSROOM EXAMPLE 5 Using the Power Rule (Squaring Twice)
Solution:
2 2 1 4( 1)x x x 2 2 1 4 4x x x
2 2 3 0x x
( 3)( 1) 0x x
3 0 or 1 0x x
3 or 1x x
Check: x = 3
2(3) 3 3 1 1
6 3 4 1
9 4 1
3 2 1 5 1FalseCheck: x = 1
2( 1) 3 1 1 1 1 0 1
1 1The solution set is { 1}.
True
Slide 10.6- 13
CLASSROOM EXAMPLE 5 Using the Power Rule (Squaring Twice) (cont’d)
Solve radical equations with indexes greater than 2.
Objective 3
Slide 10.6- 14
Solve 3 32 7 3 2.x x
Cube both sides.
Check:
3 33 32 7 3 2x x
2 7 3 2x x
9 x
3 32(9) 7 3(9) 2
3 318 7 27 2
3 325 25 True The solution set is {9}.
Slide 10.6- 15
CLASSROOM EXAMPLE 6 Using the Power Rule for a Power Greater Than 2
Solution:
Use the power rule to solve a formula for a specified variable.
Objective 4
Slide 10.6- 16
Solve the formula for R.
RZ
T
2
2( )R
ZT
2 RZ
T
2 2 or TZ R R TZ
Slide 10.6- 17
CLASSROOM EXAMPLE 7 Solving a Formula from Electronics for a Variable
Solution: