Chapter 9 Review Square Roots and Radicals Try some: Simplify these:

Chapter 9 ReviewChapter 9 Review

Square Roots and RadicalsSquare Roots and Radicals

• Try some:

• Simplify these: 25

1

0

144

16€

±5

€

±1

€

0

€

±12

€

No Solution

• More examples:144

48

196€

±12

€

±4 3

€

±13

How would you solve the equation:How would you solve the equation:xx22 = 4 = 4

(take the square root of each side!)(take the square root of each side!)

* Remember, the square root of a positive # has 2 answers! (one + and one -)

*2- 2

4

4

2

2

orx

x

x

Solving Quadratic EquationsSolving Quadratic Equations1. Solve. 3 - 5x2 = -9

-3 -3

-5x2 = -12

-5 -5

x2 = 5

12

5

122 x

5

152

5

15*4

25

60

5*5

5*12

5

12 x

2.Solve. 3(x-2)2=21 3 3

(x-2)2 = 7

7)2( 2 x

72 x

72x

More Examples!More Examples!

3. Solve. 4x2-6=42

+6 +6

4x2=48

4 4

x2 = 12

122 x

323*4 x

4. Solve. 4. Solve. 6)4(5

1 2 x

30)4( 2 x

30)4( 2 x

304 x

304x

Rationalizing the DenominatorRationalizing the Denominator

You CANNOT leave a radical in the denominator of a fraction!

No tents in the basement!!!!

(the numerator is OK)

Just multiply the top & bottom of the fraction by the radical to “rationalize” the

denominator.

Properties of Square RootsProperties of Square Roots

(a>0 and b>0)

1. Product Property –

2. Quotient Property-

baab *

Example:Example:

10210*410*440

b

a

b

a

Example:Example:

2

3

4

3

4

3

29

29

18

23

96

616

616

64

12 45

34

34

32

59

59

53

€

5( )2

55

55

25

5

€

7( )2

77

77

49

7

€

10( )2

1010

1010

100

10

2473

2743

2712

1412

3536 3356

3330

930

330

90

10528

10258

10240 2040

5440

5240

580

25147 21457

21435 2835

7435

7235

770

Find a perfect square factor of 32.Find a perfect square factor of 32.

Simplify each expression. Simplify each expression.

Product Property of Square RootsProduct Property of Square Roots

Quotient Property of Square RootsQuotient Property of Square Roots

A.A.

B.B.


Simplify each expression. Simplify each expression.


CC..

D.D.

E.E.

Simplify each expression.Simplify each expression.

F.F.

Find a perfect square factor of 48.Find a perfect square factor of 48.



Simplify.Simplify.

Simplify each expression.Simplify each expression.

G.G.

H.H.



ExamplesExamples

1.

2.

3.

500 5*100 5*100 510

6*123 6*123 723 2*363

26*3 218

9

25

9

25

3

5

More Examples!More Examples!

1.

2.

3

25

3

25

3

5

3*

3*

9

35

3

35

Can’t have a tent in the Can’t have a tent in the basement!basement!

11

2

11

2

11*

11*

121

22

11

22

Simplify by rationalizing the denominator. Simplify by rationalizing the denominator.

Multiply by a form of 1.Multiply by a form of 1.

= 2= 2

Simplify the expression. Simplify the expression.


Simplify by rationalizing the denominator.Simplify by rationalizing the denominator.


Simplify by rationalizing the denominator.Simplify by rationalizing the denominator.


1.1. Estimate to the nearest tenth. Estimate to the nearest tenth. 6.76.7Simplify each expression.Simplify each expression.

2. 2.

3.3.

4.4.

5.5.

Solving Quadratic Equations using the Quadratic Solving Quadratic Equations using the Quadratic FormulaFormula

General equation of a General equation of a quadratic:quadratic:

02 cbxax

Quadratic Quadratic Formula:Formula:

€

x =−b ± b2 − 4ac

2a

Notice where the letters come from for the formula

We use the quadratic formula when something can not be factored. However, it also works We use the quadratic formula when something can not be factored. However, it also works for factorable quadratic equations as well.for factorable quadratic equations as well.

You must get equation You must get equation equal to zero before you equal to zero before you

determine a, b, and c.determine a, b, and c.

2 6 5x x Ex. 1Ex. 1 Solve.Solve.

2 5 6 0x x 5

6

1

c

b

a

2( 5) ( 5) 4(1)

(1)

( )

2

6x

5 25 24

2x

5 1

2x

5 1

2x

3,2x

2 2 8 0x x Ex. 2Ex. 2 Solve.Solve.

2

8

1

c

b

a

2( 2) ( 2 (1)

(1)

() 4 )

2

8x

4, 2x

2 14 49 0x x Ex. 3Ex. 3 Solve.Solve.

14

4

1

9

b

a

c

2 (1)

(1

(14) (14) ( )

)

4

2

49x

7x

218 10 0x x Ex. 4Ex. 4 Solve.Solve.

10

18

1b

a

c

2 (14 8)

(

(1) (1)

82 1 )

(10)x

no solution

Ex. 5Ex. 5 Suppose a football player kicks a ball and gives it an Suppose a football player kicks a ball and gives it an initial upward velocity of 47ft/s. The starting height of initial upward velocity of 47ft/s. The starting height of the football is 3ft. If no one catches the football, how the football is 3ft. If no one catches the football, how

long will it be in the air?long will it be in the air?

NOTENOTE: For your homework tomorrow night, DO NOT answer the : For your homework tomorrow night, DO NOT answer the question. I want you to draw a sketch of what you think this picture question. I want you to draw a sketch of what you think this picture

would look like on a graph. Tell me what the x-axis symbolizes, what the would look like on a graph. Tell me what the x-axis symbolizes, what the y-axis symbolizes, and what the zeros represent in the problem.y-axis symbolizes, and what the zeros represent in the problem.

What does the x-axis stand for?What does the x-axis stand for?

What does the y-axis stand for?What does the y-axis stand for?

What do the zeros represent?What do the zeros represent?

Time the ball is in the airTime the ball is in the air

Height of the ballHeight of the ball

The amount of time it takes the ball The amount of time it takes the ball to hit the ground.to hit the ground.

01382 xx

13

8

1

c

b

a

a

acbbx

2

42

12

131488 2 x

2

128 x

3412

32 2

328 x

34x

Before you can find a,b, and Before you can find a,b, and c, you must get equation = c, you must get equation =

to 0.to 0.

Type what is under the Type what is under the radical exactly as written radical exactly as written

on your calculator.on your calculator.

Simplify the radicalSimplify the radical

Divide by the denominator, if you canDivide by the denominator, if you can

Roots:Roots:

053 2 x

5

0

3

c

b

a

a

acbbx

2

42

32

53400 2 x

6

600 x

6

152x

15460

152

3

15x

Simplify the radicalSimplify the radical

Divide by the denominator, if you Divide by the denominator, if you can. Since I can’t, divide by GCFcan. Since I can’t, divide by GCF

RootsRoots::

Solve the quadratic equation by using the Solve the quadratic equation by using the quadratic formula and leave answers in quadratic formula and leave answers in

simplest radical form.simplest radical form.

Solve the quadratic equation by Solve the quadratic equation by using the quadratic formula and using the quadratic formula and

round answers to the nearest tenth.round answers to the nearest tenth.

029122 xx

29

12

1

c

b

a

a

acbbx

2

42

12

29141212 2 x

2

2812 x

Since we are solving to the nearest Since we are solving to the nearest tenth, we do not need to simplify tenth, we do not need to simplify

the radical!the radical!

2

2812 x

2

2812 x

64.8x6.8x

35.3x4.3x

Be careful when typing this into Be careful when typing this into your calculator. I recommend that your calculator. I recommend that you type in the numerator then hit you type in the numerator then hit enter, then divide by denominator!enter, then divide by denominator!

Roots:Roots:

xx 1082

8

10

1

c

b

a

a

acbbx

2

42

12

8141010 2 x

2

6810 x

x10 x10

08102 xxSolve the quadratic equation by Solve the quadratic equation by using the quadratic formula and using the quadratic formula and

round answers to the nearest tenth.round answers to the nearest tenth.

2

6810x

2

6810 x

12.9x1.9x

87.x9.x

Roots:Roots:

Solve the quadratic equation by Solve the quadratic equation by using the quadratic formula and using the quadratic formula and

round answers to the nearest tenth.round answers to the nearest tenth.€

3x x − 2( ) =1

163 2 xx1 1

0163 2 xx

1

6

3

c

b

a

a

acbbx

2

42

32

13466 2 x

6

486x

6

486 x

6

486 x

15.2x1.2x

15.x2.x

RootsRoots

Solve the quadratic equation by using Solve the quadratic equation by using the quadratic formula and leave the quadratic formula and leave answers in simplest radical form.answers in simplest radical form.

xx 412 2 x4 x4

0142 2 xx

1

4

2

c

b

a

a

acbbx

2

42

22

12444 2 x

4

84x

248

224

224x

Divide by the denominator, if you Divide by the denominator, if you can. Since I can’t, divide by GCFcan. Since I can’t, divide by GCF

2

22 x

Roots:Roots:

Documents

Chapter 9 Review Square Roots and Radicals Try some: Simplify these: