For Educational Use Only © 2010 12.4 Completing the Square Brian Preston Algebra 1 2009-2010.

For Educational Use Only © 2010

12.4 Completing the Square

Brian PrestonAlgebra 1 2009-

2010

For Educational Use Only © 2010?

Real World Application

How far away from the end of the board can you dive into the water?

10 ft


Lesson Objectives

1) Solve a quadratic equation by completing the square.

2) Choose a method for solving a quadratic equation.


Example

1) What are the ways to solve a quadratic equation (ax2 + bx + c = 0)?

Square Roots

Graphing

Factoring

Quadratic Formula

Completing the Square


Definition


x2 + bx + = x +b

2b

2)( 2 )( 2

To make this more manipulative we will add a term.

Factoring the left terms will get you the right terms.


Definition


x2 + bx + = x +b

2b

2)( 2 )( 2

To make this more manipulative we will add a term.

Perfect SquareThere has to be a 1 to do this process.

1


(x + 3)2

Definition

Completing the Square using algebra tiles.

x2 + 6x

Make a square

x2 x x x x

xxx

111

111

111

x2 + 6x + 9 =+ ( )26

26

x + 3

x x


Example

Find the term that should be added to the expression to create

a perfect square trinomial.

2) x2 – 8x +– 8

2)( 2

or 16

1


Example

3) x2 + 0.4x +0.4

2)( 2

or 0.04

Find the term that should be added to the expression to create

a perfect square trinomial.

1


– 0.44x2 + 2.61x + 10

4) The path of a diver diving from a 10-foot high diving board is h = – 0.44x2 + 2.61x + 10 where h is the height of the diver above water (in feet) & x is the horizontal distance (in feet) from the end of the board. How far away from the end of the board will the diver enter the water?

h = – 0.44x2 + 2.61x + 10

far

1010h = – 0.44x2 + 2.61x + 10

0 =


h

farfar


2.61–0.44 2.612.61–0.44–0.44

b (2.61) b a

1010

2 – 4(–0.44)

(–0.44) (10)

Solve – 0.44x2 + 2.61x + 10 = 0

ax2 + bx + c = 0a = b =

2.61– 0.44

c =

10

x = b2 – 4ac– b ±

2a

x = – ±

2

(2.61) c a

Standard form


4)


4)x =

6.8121 x =

+ 17.6±

– 0.88

– 2.61

24.4…x =

±

– 0.88

–2.61 4.94…±

– 0.88

–2.61=


(2.61) 2 – 4

(–0.44)

(–0.44) (10) – ±

2

(2.61)


x =

x =

x =

4.94…±

– 0.88

– 2.61

4.94…+

– 0.88

– 2.61=

4.94…–

– 0.88

– 2.61 =

– 2.65

8.58

x = 8.58 feet

4)




How far away from the end of the board can you dive into the water?

10 ft

8.58ft

?


Example

5) x2 + 10x =

Solve the equation by completing the square.

241


+ ( )210

2(x + 5)2

DefinitionCompleting the Square using algebra tiles.

x2 + 10x

Make a square

x2 x x x x

xxx

111

111

111

x2 + 10x + 25 =10

x + 5

x x x x x xx

1 1 1x

1 1 1

11111

11111


Example

5) x2+10x =10

2)( 2


+ 2410

2)( 2+

x2 + 10x =+ 24 +52 52

x + )( 25 = 49

5

x + 5 = 7±

10


– 5– 5 – 5

Example

5)


x + 5 = 7±

x – 5= 7±

x – 5= + 7

x – 5= – 7= 2= – 12

x = 2,– 12


– 4– 4

Example

6) x2 – 6x +


4 = 20– 4

x2 – 6x = 16

1


Definition


x2 – 6x

Make a square

x2 x x x x

xxx

111

111

111

x2 – 6x + 9+ ( )2– 6

2– 6

x x


(x – 3)2

Definition


x2 – 6x

Make a square

x2 x x x

xxx

1

1

1

1

1

1

1

1

1

x2 – 6x + 9 =

x – 3


–3–3–

Example

6) x2 – 6x =-6

2)( 2


+ 16-6

2)( 2+

x2 6x =+ 16 + –3

x – )( 23 = 25

x – 3 = 5±

– 6

)2( )2(


+ 3+ 3 + 3

Example

6)


x – 3 = 5±

x 3= 5±

x 3= + 5

x 3= – 5= 8= – 2

x = 8,– 2


+ 3+ 3

Example

7) x2 – x –


3 = 0+ 3

x2 – x = 3

1


1

2

-1

2-1

2–

Example

7) x2 – x =-1

2)( 2


+ 3-1

2)( 2+

x2 1x =+ 3 + )2

x – )( 2 =

x – = ±

1

1

2

()2(-1

2

– 1

13

413

2


x – = ±1

2

13

2

++ +

Example

7)


x = ±

1

21

2

1

213

2

1

2


x – = ±1

2

13

2

+ +

Example

7)


x = ±

1

21

2

1

213

2


2 2 2

+ 13+ 13

Example

8) 2x2 – 8x –


13 = 7+ 13

2x2 – 8x = 20

1

2 2

x2 – 4x = 10


+ ( )2

Definition


x2 – 4x

Make a square

x2 x x x

xx 1

111

x2 – 4x + 4– 4

2– 4

x


(x – 2)2

Definition


x2 – 4x

Make a square

x2 x x

xx

1

1

1

1

x2 – 4x + 4 =

x – 2


4x

–2–2–

Example

8) x2 – =-4

2)( 2


+ 10-4

2)( 2+

x2 4x =+ 10 + –2

x – )( 22 = 14

x – 2 = 14±

)2( )2(

– 4


+ 2+ 2 + 2

Example

8)


x – 2 = ±

x 2= ±

14

14


+ 2 + 2

Example

8)


x – 2 = ±

x 2= ±

14

14


4 4 4

+ 11+ 11

Example

9) 4x2 + 4x –


11 = 0+ 11

4x2 + 4x = 11

1

4 4

x2 + 1x =11

4


11

4

1

21

21

2

1

2+

Example

9) x2 + x =1

2)( 2


+1

2)( 2+

x2 1x =+ + )2

x + )( 2 =

x + =

1

1

2

()2(

1 11

4

3

3±


–––

x + =1

2

Example

9)


x =

1

21

2

1

2

1

2

3±

3±–


x + =1

2

– –

Example

9)


x =

1

21

2

1

2

3±

3±–


3 3

+ 15+ 1510) 3x2 – 15 = 0

Choose a method to solve the quadratic equation.

Example

+ 15

3x2 = 153

x2 = 5 x = 5±


11) x2 + 12x + 20 = 01Factors of 20

1 20+ 12

+12=2 10

Choose a method to solve.

Standard form

1 + 20 21=

20 – 1 19=

Example

4 5


8

11) x2 + 12x + 20 = 01

2 + 10 +12=


1 20

Factors of 20

2 + 10 12=

10 – 2 =

Example

2 10

4 5


1x 1x1x 1x+10+10

+10 + 2

11) x2 + 12x + 20 = 01

+2+2

2 + 10102(

+12=


1 20

Factors of 20

)( )

1x2

= 0

Example

2 10

4 5


– 10– 10

(x + 10)(x + 2)(x + 10)(x + 2)

– 2– 2x + 10


11) = 0

= 0= 0 x + 2

x

– 10

= – 10

( ) ( )

x

– 2

= – 2

Example


– 26– 26

2 – 4

222

(1)

(1) (– 26) a (2)

12) Solve x2 + 2x – 26 = 01ax2 + bx + c = 0

a = b =

2

1

1

c =

– 26

x = b2 – 4ac– b ±

2a

x = – ±

2

(2) b b c a

11

Standard form

Example


(1)

(1) (– 26) (2) 12)x =

2 – 4– ±

2

(2)

4 x =

+ 104±

2

– 2

108x =

±

2

– 2 10.39±

2

– 2 =

Example


x =

x =

x =

10.39±

2

– 2

10.39+

2

– 2=

10.39–

2

– 2 =

8.39

2= 4.196

– 12.39

2=

– 6.196

x = 4.196 & – 6.196

12)

Example


Key Points & Don’t Forget

1) Don’t forget the negative signs.

2) Add to both sides of the equation

3) When taking the square root of an equation you will get a ± answer.

b

2)( 2


The Assignment

pg. 624-626 #’s 8-58 even, 62-65


Bibliography

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For Educational Use Only © 2010 12.4 Completing the Square Brian Preston Algebra 1 2009-2010.

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Transcript of For Educational Use Only © 2010 12.4 Completing the Square Brian Preston Algebra 1 2009-2010.