[email protected] mth55_lec-45_7-6a_radical_equations.ppt 1 bruce mayer, pe chabot college...
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§7.6 Radical§7.6 RadicalEquationsEquations
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §7.5 → Rational Exponents
Any QUESTIONS About HomeWork• §7.5 → HW-34
7.5 MTH 55
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt3
Bruce Mayer, PE Chabot College Mathematics
Radical EquationsRadical Equations
A Radical Equation is an equation in which at least one variable appears in a radicand.
Some Examples:
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt4
Bruce Mayer, PE Chabot College Mathematics
Power Rule vs Radical EqnsPower 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
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt5
Bruce Mayer, PE Chabot College Mathematics
Caveat PowerRule → CheckCaveat PowerRule → Check
CAUTIONCAUTION 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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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt6
Bruce Mayer, PE Chabot College Mathematics
Power RulePower Rule
The Power Rule Provides a Crucial Tool for solving Radical Equations.
Recall the Power Rule
If a = b, then an = bn for any natural-number exponent n
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt7
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve by PwrRule Solve by PwrRule
Solve Radical Equations:a) b)12y 3 4x
SOLUTIONa) b) 2
212y
144y
144 12
12 12
Check
True
3 33 4x
64x
Check3 64 4
4 4 True
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt8
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION
4 60x x
4 60x x
2 2
4 60x x
224 60x x
16 60x x
15 60x
4x
Check
4 60x x
4 4 4 60
4 2 64
8 8 4 Satisfies the
original Eqn, so 4 is verified as a Solution
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt9
Bruce Mayer, PE Chabot College Mathematics
Solving Radical EquationsSolving 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
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION
5 7x x
5 7x x
225 7x x
2 10 25 7x x x 2 11 25 7x x 2 11 18 0x x
( 2)( 9) 0x x 2 0 or 9 0x x
2x 9x
Square both sides.
Use FOIL or Formula.
Subtract x from both sides.
Factor.
Use the zero-products theorem.
Subtract 7 from both sides.
The TENTATIVE Solutions
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt11
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve 5 7x x
Check BOTH Tentative Solutions2x
2 5 2 7
3 9
3 3 False.
9x
9 5 9 7
4 16
4 4 True.
Because 2 does not check, it is an extraneous solution. The only soln is 9
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt12
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION Check
4 6x
4 6x
2x
2 2
2x
4x
4 6x
4 4 6
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}
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt13
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION Check
So 13 checks. The solution set is {13}
4 3 3 5.x
4 3 3 5x 4 3 2x
444 3 2x
3 16x
13x
4 3 3 5x
4 13 3 3 5
4 16 3 5
2 3 5
5 5
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt14
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLN
Check
So 9 checks. The solution set is {9}
3 9m 6m
2 26m 36m
Using the Power Rule
Isolate the variable radical
3 9m
36 3 93 9m
6 3 9
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt15
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLN
5 1x x 5 1x x
1 5x x
221 5x x
2 2 1 5x x x 2 3 4 0x x
0( 4)( 1)x x
Isolate the variable radical
Sq Both Sides to Remove Radical
(x−1)2 ≠ x2 −12
04 0 or 1x x 14 or x x
Apply Zero-Products
Tentative Solutions
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt16
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
5 1x x Check BOTH Tentative Solutions
5 1x x
4 5 1 9 13+1
4
4
5 1x x
1 5 1 4 12+1
−1
−1 In this Case 4 checks while −1 does
NOT. The solution set is {4}
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt17
Bruce Mayer, PE Chabot College Mathematics
Example Example Solve Solve
SOLUTION
3 23 4x
33 23
3 4x
3 4 8x
3 4x
4 / 3x
3 2 03 4x
CHECK
02433 x
0243
43
?
3
0244?
3
028?
3
022
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt18
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard 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
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt19
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
LifeExpectancy
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt20
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt21
Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
-4
-3
-2
-1
0
1
2
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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
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[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt22
Bruce Mayer, PE Chabot College Mathematics
-3
-2
-1
0
1
2
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4
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-3 -2 -1 0 1 2 3 4 5
M55_§JBerland_Graphs_0806.xls -5
-4
-3
-2
-1
0
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-10 -8 -6 -4 -2 0 2 4 6 8 10
M55_§JBerland_Graphs_0806.xls
x
y