[email protected] MTH55_Lec-45_7-6a_Radical_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-45_7-6a_Radical_Equations.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.6 Radical§7.6 RadicalEquationsEquations



Review §Review §

Any QUESTIONS About• §7.5 → Rational Exponents

Any QUESTIONS About HomeWork• §7.5 → HW-34

7.5 MTH 55



Radical EquationsRadical Equations

A Radical Equation is an equation in which at least one variable appears in a radicand.

Some Examples:



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



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



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



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



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



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




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



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




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}




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




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




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




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}




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



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



All Done for TodayAll Done for Today

LifeExpectancy



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



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

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls -5

-4

-3

-2

-1

0

1

2

3

4

5

-10 -8 -6 -4 -2 0 2 4 6 8 10

M55_§JBerland_Graphs_0806.xls

x

y

Documents

[email protected] MTH55_Lec-45_7-6a_Radical_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical