Warm Up

y = 8x2 – 16x -10

= roots

a = 8, b = – 16, c = -10

Axis of symmetry = -b 2a

x = -(-16) 2(8)

= 1

y = 8(1)2 – 16(1) -10 = -18

Vertex= minimum

a > 0 parabola opens up

(1, -18)

y-intercept

Axi

s of

sym

met

ry

y = 4x2 – 16x + 15

= roots

a = 4, b = – 16, c = 15

Axis of symmetry = -b 2a

x = -(-16) 2(4)

= 2

y = 4(2)2 – 16(2) +15 = -1

Vertex= minimum

a > 0 parabola opens up (2, -1)

y-intercept

To plot one more point: Select any x and solve for y

Ex: when x = 1,y = 4(1)2 – 16(1) + 15 =3 (1,3)

(1,3) (3,3)

0 = 32t – 16t2

h = 32t – 16t2 h = – 16t2 + 32t

a < 0 parabola opens down

Vertex= maximum

(1, 16)

At what time will the ball be 8 meters in the air?

Axi

s of

sym

met

ry

8 = – 16t2 + 32t

0 = – 16t2 + 32t -8

0 = -8(2t2 - 4t – 1)Use the quadratic formula to find t.a = 2, b = -4, c = -1

0 05x3x2x3 2

1x 1x

-5- 5 2+ 2

x2

( )( )

-10

-3

x2

5x2 2

0x15x9x6 23

x3

Set the factors equal to zero and solve.

05x2 01x 0x3

25

x

5x2

1x

You must keep the greatest

monomial factor that is

pulled out beforebefore using the X figure!

Can you factor out a greatest monomial

factor?

0x33

More factoring and solving. Solve. x15x4x9x2 323

8-5 Factoring Differences of Squares

Algebra 1 Glencoe McGraw-Hill Linda Stamper

Difference of Two Squares

22 bababa

factors

product

9x2

Recognizing a difference of two squares may help you to factor - notice the sum and difference pattern. 22 3x 3x3x

64x81 2 222 8x9 8x98x9

No middle term – check if first and last terms are squares. Sign is negative.

Check using FOIL!

Factor.

5x5x 25x2 6x6x 36x2

2x22x2 4x4 2 4x34x3 16x9 2

4x24x2 16x4 2

5x45x4 25x16 2 Sign must be negative!

16x49 2 prime

Example 1 100y9 2

10y310y3

Check using FOIL!

Factor.

Example 2 81m64 2

9m89m8

Example 3

36m49 2 6m76m7

Example 4 9n1212

3n113n11

Example 5 144y16 2

12y412y4

Example 6 25x36 2

5x65x6

Remember to factor completely.

Write problem. 100x25 2

No middle term – check if first and last terms are squares.

2x2x25

Factor – must use parentheses.

Check using FOIL!

Factor out the GMF. 4x25 2

100x25 2 222 10x5

10x510x5 2x5 2x5 2x2x25

Sometimes you may need to apply several different factoring techniques.

15x5x15x5 23

Group terms with common factors.

Factor each grouping.Factor the common binomial factor.

Check – Multiply the factors together using FOIL.

The problem.

Factor out the GMF.

3xx3x5 23

3x3xx5 23

1x31xx5 22

3x1x5 2

Factor the difference of squares.

3x1x1x5

Example 7 1y4

1y1y 22

1y1y1y2

Factor.Example 8

44 b4a4

44 ba4

Example 9

81x4

9x9x 22

Example 10

2222 baba4

bababa4 22

9x9x9x2

120x24x30x6 23 20x4x5x6 23

20x5x4x6 23

4x54xx6 22

5x4x6 2

5x2x2x6

Use factoring to solve the equation. Remember to set each factor equal to zero and then solve!

081y16 2

Example 11 Example 12

169

x2

0x4x9 3

Example 13 Example 14 120d24d30d6 23

Use factoring to solve the equation. Remember to set each factor equal to zero and then solve!

081y16 2

09y4 or 09y4

49

y

9y4

49

y

Example 11 Example 12

09y49y4

9y4

169

x2

043

x or 043

x

4

3x

4

3x

043

x43

x

0169

x2

Use factoring to solve the equation. Remember to set each factor equal to zero and then solve!

0x4x9 3

02x3 or 02x3or0x

32

x

2x3

32

x

Example 13

02x32x3x

2x3

04x9x 2

0x

Use factoring to solve the equation. Remember to set each factor equal to zero and then solve!

Example 14

120d24d30d6 23

020d4d5d6 23

020d5d4d6 23

04d54dd6 22

05d4d6 2

05d2d2d6

05d or 02dor02dor06 2d 5d 2d 06

0120d24d30d6 23

8-A11 Pages 451 # 11–30.