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Transcript 
Chabot Mathematics
§7.6 Radical
Equations
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Chabot College Mathematics
1
Bruce Mayer, PE
[email protected] • 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-35
Chabot College Mathematics
2
Bruce Mayer, PE
[email protected] • 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
3
Bruce Mayer, PE
[email protected] • 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
4
Bruce Mayer, PE
[email protected] • 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
Chabot College Mathematics
5
Bruce Mayer, PE
[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Power Rule
 The Power Rule Provides a Crucial Tool
for solving Radical Equations.
 Recall the Power Rule
n
a
n
b
If a = b, then = for any
natural-number exponent n
Chabot College Mathematics
6
Bruce Mayer, PE
[email protected] • 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
7
  4 
3
x  64
Check
144  12
12  12
3
3
True
64  4
4   4
True
Bruce Mayer, PE
[email protected] • 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
8
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
[email protected] • 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
9
Bruce Mayer, PE
[email protected] • 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
10
Square both sides.
Use FOIL.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-products theorem.
The TENTATIVE Solutions
Bruce Mayer, PE
[email protected] • 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
11
Bruce Mayer, PE
[email protected] • 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
12
Bruce Mayer, PE
[email protected] • 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
13
Bruce Mayer, PE
[email protected] • 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
Chabot College Mathematics
14

 So 9 checks. The
solution set is {9}
Bruce Mayer, PE
[email protected] • 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
15
Apply Zero-Products
Tentative Solutions
Bruce Mayer, PE
[email protected] • 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
16
Bruce Mayer, PE
[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt
Example  Solve 3 3x  4  2  0.
 SOLUTION
3 3x  4  2  0

3 3x  4  2
3 3 x  4 3  23

3x  4  8
3x  4
x  4/3
Chabot College Mathematics
17
 Because both
sides were raised
to an odd power,
it is not essential
that we check the
answer
• Recall that
Negative numbers
can be valid Solns
to Odd-Index
Equations
Bruce Mayer, PE
[email protected] • 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
18
Bruce Mayer, PE
[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt
All Done for Today
Life
Expectancy
Chabot College Mathematics
19
Bruce Mayer, PE
[email protected] • 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
[email protected]
–
Chabot College Mathematics
20
Bruce Mayer, PE
[email protected] • 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
21
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
-6
Bruce Mayer, PE
[email protected] • 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
22
-4
M55_§JBerland_Graphs_0806.xls
-5
Bruce Mayer, PE
[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt
8
10
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