For Educational Use Only © 2010 10.6 Factoring ax 2 + bx + c Brian Preston Algebra 1 2009-2010.

For Educational Use Only © 2010

10.6 Factoring ax2 + bx + c

Brian PrestonAlgebra 1 2009-

2010


shirt

Real World Application

How long until you catch the shirt?

70 ft.

6 ft.


Lesson Objectives

1) Factor a quadratic expression of the form ax2 + bx + c.

2) Solve quadratic equations by factoring.


1) x2 – 2x – 81Factors of 8

1 8– 2

-2=

2 4

Factor the trinomial.

Review

1 + 8 9=

8 – 1 7=


1) x2 – 2x – 81

2 + – 4 -2=


1 8

2 4

Factors of 8

Review

2 + 4 6=

4 – 2 2= (–1) to all


1) x2 – 2x – 8

– 4– 41x1x

1

+2+2

2 + – 4– 42( – 4 + 2

-2=


1 8

2 4

Factors of 8

)(1x )

1x2

1x

Review


2) A mascot shoots t-shirts from a “T-shirt cannon” into the crowd. The T-shirts are released from a height of 6 feet with an initial upward velocity of 72 feet per second. If you catch a T-shirt at your seat 70 feet above the court, how long was it in the air before you caught it?

72727070

+= – 16t2

longlong

6

long

6


72th v s70 6

upward

+


– 70– 70

– 4


2) – 16t2 + 72t + 6 = 70

Solve the equation by factoring.

– 70

=+ 72t– 16t2 0– 64

Standard form

=– 18t4t2 + 16 0– 4 – 4 – 4 – 4 – 4 – 4


– 18– 18– 18444 1616

a

ax2 + bx + c = 0

(4)

(4) (16) (– 18)

2) 4t2 – 18t + 16 = 0

a = b =

– 184

c =

16

t = b2 – 4ac– b ±

2a

t = 2 – 4– ±

2

(– 18) b b c a

Standard form



(4) 2)

324 t =

– 256±

8

18

68t =

±

8

18

(4) (16) (–18)

t = 2 – 4– ±

2(–18)



t =

t =

t =

8.24±

8

18

8.24+

8

18 =

8.24–

8

18 =

3.28

1.22

t = 3.28 seconds

2)


Way down

Way up


shirt


How long until you catch the shirt?

70 ft.

6 ft.3.28 seconds


Example

3) 3x2 – 4x - 7Factors of 21

1 21– 4

-4=

3 7

Now the numbers are getting bigger.

1 + 21 22=

21 – 1 20=


3 + – 7

Example

3) 3x2 – 4x - 7

-4=


1 21

3 7

Factors of 21

3 + 7 10=

7 – 3 4= (–1) to all


3 + – 7– 73( – 7 + 1

Example

3) 3x2 – 4x - 7

-4=


1 21

3 7

Factors of 21

+3

– 73x )(1x )

3x2

3x

+3

– 71x


Rule

x2 – 2x – 8

One (-) & One (+)

Patterns for factoring trinomials.

(x + 2)(x - 4)

x2 + 6x + 8(x + 2)(x + 4)

x2 – 6x + 8(x – 2)(x – 4)

Two (+)Two (-)

x2 + 2x – 8(x + 4)(x - 2)


1 + 10

Example

4) 2x2 + 11x + 5Factors of 10

1 10+ 11

+11=

2 5


1 + 10 11=

10 – 1 9=


+1+1

+102x2x1 + 1010

4) 2x2 + 11x + 5

1(

Example

+11=


1 10

2 5

Factors of 10

)(1x )

2x2

Switch


+10+10

+1+12x2x1 + 10

4) 2x2 + 11x + 5

( + 1 + 5

Example

+11=


1 10

2 5

Factors of 10

)(1x )1x


Example

5) 6a2 – 19a + 15Factors of 90

1 90– 19

-19=2 45


3 305 186 159 10

1 + 90 91=

90 – 1 89=


Example

5) 6a2 – 19a + 15Factors of 90

1 90

-19=2 45


3 305 186 159 10

2 + 45 47=

45 – 2 43=


Example

5) 6a2 – 19a + 15Factors of 90

1 90

-19=2 45


3 305 186 159 10

3 + 30 33=

30 – 3 27=


Example

5) 6a2 – 19a + 15Factors of 90

1 90

-19=2 45


3 305 186 159 10

5 + 18 23=

18 – 5 13=


Example

5) 6a2 – 19a + 15Factors of 90

1 90

-19=2 45


3 305 186 159 10

6 + 15 21=

15 – 6 9=


5) 6a2 – 19a + 15

– 9 + – 10

Example

-19=

Factor the trinomial.Factors of 90

1 902 453 305 186 159 10

9 + 10 19=

10 – 9 1= (–1) to all


-9

–10

-9

–102a2a3a3a

5) 6a2 – 19a + 15

– 9 + – 10– 10– 9( – 5 – 3

Example

-19=


)( )

6a2 1 902 453 305 186 159 10


Example

6) 6x2 – 2x – 8Factors of 48

1 48– 2

-2=2 24


3 164 126 8

1 + 48 49=

48 – 1 47=


6) 6x2 – 2x – 8

Example

Factors of 48

1 48

-2=2 24


3 164 126 8

2 + 24 26=

24 – 2 22=


6) 6x2 – 2x – 8

Example

Factors of 48

1 48

-2=2 24


3 164 126 8

3 + 16 19=

16 – 3 13=


6) 6x2 – 2x – 8

Example

Factors of 48

1 48

-2=2 24


3 164 126 8

4 + 12 16=

12 – 4 8=


6) 6x2 – 2x – 8

6 + – 8

Example

-2=


1 482 243 164 126 8

6 + 8 14=

8 – 6 2= (–1) to all


+6

–8+6

–83x3x 2x2x

6) 6x2 – 2x – 8

6 + – 8– 86( – 4 + 2

Example

-2=


)( )

6x2 1 482 243 164 126 8


Example

7) 8t2 – 4t – 5Factors of 40

1 40– 4

-4= 2 20


4 10

5 81 + 40 41=

40 – 1 39=


7) 8t2 – 4t – 5

Example

Factors of 40

1 40

-4= 2 20


4 10

5 82 + 20 22=

20 – 2 18=


7) 8t2 – 4t – 5

Example

Factors of 40

1 40

-4= 2 20


4 10

5 84 + 10 14=

10 – 4 6=


7) 8t2 – 4t – 5

Example

Factors of 40

1 40

-4= 2 20


4 10

5 8Not Factorable

5 + 8 13=

8 – 5 3=


Example

8) 2z2 – 17z – 19 = 0Factors of 38

1 38– 17

-17=

2 19


1 + 38 39=

38 – 1 37=


8) 2z2 – 17z – 19 = 0

2 + – 19

Example

-17=


1 38

2 19

Factors of 38

2 + 19 21=

19 – 2 17= (–1) to all


2z 1z2z 1z

+2+2

–19–19

8) 2z2 – 17z – 19 = 0

2 + – 19– 192( – 19 + 1

Example

-17=


1 38

2 19

Factors of 38

)( )=0

2z2


2

+ 192z – 19

+ 19

2z

2

– 1– 1

(2z – 19)(1z + 1)(2z – 19)(1z + 1)

ExampleSolve the equation by factoring.

8) = 0

= 0= 0 1z + 1+ 19

= 19

( ) ( )

z

– 1

= – 1+ 0 + 02 z = z = – 119

2


+ 4x+ 4x

– 2– 2

Example

9) 28x2 – 9x – 1 = – 4x + 2


– 2

=– 9x28x2 + 0– 3

Standard form

– 4x+ 4x

=– 5x28x2 – 3 0


Example

9) 28x2 – 5x – 3 = 0Factors of 84

1 84– 5

-5=2 42


3 284 216 147 12

1 + 84 85=

84 – 1 83=


Example

9) 28x2 – 5x – 3 = 0Factors of 84

1 84

-5=2 42


3 284 216 147 12

2 + 42 44=

42 – 2 40=


Example

9) 28x2 – 5x – 3 = 0Factors of 84

1 84

-5=2 42


3 284 216 147 12

3 + 28 31=

28 – 3 25=


Example

9) 28x2 – 5x – 3 = 0Factors of 84

1 84

-5=2 42


3 284 216 147 12

4 + 21 25=

21 – 4 17=


Example

9) 28x2 – 5x – 3 = 0Factors of 84

1 84

-5=2 42


3 284 216 147 12

6 + 14 20=

14 – 6 8=


9) 28x2 – 5x – 3 = 0

7 + – 12

Example

-5=


Factors of 84

1 842 423 284 216 147 12

7 + 12 19=

12 – 7 5= (–1) to all


4x4x7x7x

+7

–12

+7

–12

9) 28x2 – 5x – 3 = 0

7 + – 12– 127( – 3 + 1

Example

-5=


Factors of 84

)( )=0

28x2 1 842 423 284 216 147 12


4 4

+ 3+ 3

7

(4x + 1)(7x – 3)(7x – 3)(4x + 1)

7x – 3

7x

7

– 1– 1

ExampleSolve the equation by factoring.

9) = 0

= 0= 0 4x + 1+ 3

= 3

( ) ( )

4x

– 1

= – 1+ 0 + 07 x = x =3

7

1

4 – 4


1) Don’t forget the negative signs.

2) Make sure the expressions and equations are in standard form before factoring.

Key Points & Don’t Forget


pg. 509-510 #’s 16-44 even

The Assignment


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For Educational Use Only © 2010 10.6 Factoring ax 2 + bx + c Brian Preston Algebra 1 2009-2010.

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