Chabot Mathematics
§7.6 Radical
Equations
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Review § 7.5
MTH 55
 Any QUESTIONS About
• §7.5 → Rational Exponents
 Any QUESTIONS About HomeWork
• §7.5 → HW-28
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Radical Equations
 A Radical Equation is an equation
in which at least one variable
appears in a radicand.
 Some Examples:
4
5 x 1  4  1 and m 2
4  1 and m 2  m  9.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Power Rule vs Radical Eqns
 Power Rule for Solving Radical
Equations:
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
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Caveat PowerRule → Check
 CAUTION
 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.
 Thus the CHECK is CRITICAL
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
ReCall Exponent Power Rule
 The Power Rule Provides a Crucial Tool
for solving Radical Equations.
 Recall the Exponent Power Rule
n
a
n
b
If a = b, then = for any
natural-number exponent n
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve by PwrRule
 Solve Radical Equations:
a)
y  12
b)
 SOLUTION
a)
 y
2
 12
2
b)
3
x  4
 x
3
y  144
Check
Chabot College Mathematics
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  4 
3
x  64
Check
144  12
12  12
3
3
True
64  4
4  4
True
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve 4 x  x  60
 SOLUTION
 Check
4 x  x  60
4 x   
2
42
 x
2
x  60
 x  60
16x  x  60
15x  60
x4
Chabot College Mathematics
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4 x  x  60

2
4 4  4  60
4  2  64
88
 4 Satisfies the
original Eqn,
so 4 is verified
as a Solution
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Solving Radical Equations
1. Isolate the radical.
 If there is more than one radical term, then isolate
one of the radical terms.
2. Raise both sides of the equation to the same
power as the root index.
3. If all radicals have been eliminated, then solve.
If a radical term remains, then isolate that
radical term and raise both sides to the same
power as its root index.
4. Check each solution. Any apparent solution
that does not check is an extraneous solution
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve x  5  x  7
 SOLUTION
x 5  x 7
 x  5
2


x7

2
x 2  10 x  25  x  7
x 2  11x  25  7
x 2  11x  18  0
( x  2)( x  9)  0
x2  0
or x  9  0
x2
x 9
Chabot College Mathematics
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Square both sides.
Use FOIL or Formula.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-products theorem.
The TENTATIVE Solutions
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve x  5  x  7
 Check BOTH Tentative Solutions
x2
25  27
95  97
4  16
3  9
3  3
x 9
False.
44
True.
 Because 2 does not check, it is an
extraneous solution. The only soln is 9
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve x  5  x  7
 What Produced the Extraneous Solution?
 At this step we Squared a NEGATIVE
Number withOUT Knowing it…
 x  5
2


x7

2
Square both sides.
 If x = 2, then (x−5) = −3
• So Squaring (x−5) is the SAME as
Squaring −3; we just didn’t know it
– Thus 2 is a solution to x 2  11x  18  0
– But NOT a solution to
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x  x 7 5  0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve
 SOLUTION
x  4  6
 Check
x  4  6
x  4  6
x  2
4  4  6
 x
2
  2 
x4
2
2  4  6
2  6

 This tentative solution x=4 does not
check, so it is an extraneous solution.
The equation has no solution; the
solution set is {Ø}
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve
 SOLUTION
4
4
x3  2
x3

4
 24
x  3  3  5.
 Check
x3 3 5
4

4
4
4
x3 3 5
13  3  3  5
4
16  3  5
x  3  16
23  5
x  13
55

 So 13 checks. The solution set is {13}
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve
 SOLN
m 39
m 6
 m
2
6
m  36
2
m 39
Isolate the variable radical
Using the Power Rule
 Check
m 39
36  3 9
639
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
 So 36 checks. The
solution set is {36}
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve x  x 5  1
x  x 5  1
 SOLN
x  1  x 5
Isolate the variable radical
 x1   x5 
2
x  2x  1  x  5
2
2
Sq Both Sides to
Remove Radical
(x−1)2 ≠ x2 −12
x  3x  4  0
2
( x  4)( x  1)  0
x  4  0 or x  1  0
x  4 or x  1
Chabot College Mathematics
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Apply Zero-Products
Tentative Solutions
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve x  x 5  1
 Check BOTH Tentative Solutions
x  x 5  1
4
4
4 5  1
9 1
3+1
x  x 5  1
−1

−1
15  1
4 1
2+1

 In this Case 4 checks while −1 does
NOT. The solution set is {4}
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve 3 3x  4  2  0.
 SOLUTION
 CHECK

3 3x  4  2
3 3 x  4 3  23

3x  4  8
3x  4
x  4/3
Chabot College Mathematics
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3x  4  2  0
3
3 3x  4  2  0
3
?
4
3   4  2  0
3
3
?
4 4  20
3
?
8  20
22  0

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
WhiteBoard Work
 Problems From §7.6 Exercise Set
• 20, 26, 30, 46, 56

Remember, Raising
Both Sides of Eqn
to an EVEN Power
can introduce
EXTRANEOUS
Solutions
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
All Done for Today
Life
Expectancy
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
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5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
4
5
6
5
5
y
4
4
3
3
2
2
1
1
0
-10
-8
-6
-4
-2
-2
-1
0
2
4
6
-1
0
-3
x
0
1
2
3
4
5
-2
-1
-3
-2
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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M55_§JBerland_Graphs_0806.xls
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-45_7-6a_Radical_Equations.ppt
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