Bellringer part two

• Simplify• (m – 4)2.

• (5n + 3)2.

Determine the pattern1

4

9

16

25

36

…

= 12

= 22

= 32

= 42

= 52

= 62

These are perfect squares!

You should be able to list at least the first 15 perfect squares in 30 seconds…

GO!!!• Perfect squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

How far did you get?

Perfect Square TrinomialAx2 + Bx + C

• Clue 1: A & C are positive, perfect squares.

• Clue 2: B is the square root of A times the square root of C, doubled.

If these two things are true, the trinomial is a Perfect Square Trinomial and can be

factored as (x + y)2 or (x – y)2.

General Form of Perfect Square Trinomials

• x2 + 2xy + y2 = (x + y)2

or• x2 – 2xy + y2 = (x - y)2

• Note: When factoring, the sign in the binomial is the same as the sign

of B in the trinomial.

Just watch and think.• Ex) x2 + 12x + 36• What’s the square root of

A? of C?• Multiply these and double.

Does it = B?• Then it’s a Perfect Square

Trinomial!

• Solution: (x + 6)2

• Ex) 16a2 – 56a + 49• Square root of A? of

C?• Multiply and double…• = B?

• Solution: (4a – 7) 2

Ex. 1: Determine whether each trinomial is a perfect square trinomial.

If so, factor it.1. y² + 8y + 162. 9y² - 30y + 10

Example 2: Factoring perfect square trinomials.

• 1) x2 + 8x + 16 2) 9n2 + 48n + 64

• 3) 4z2 – 36z + 81 4) 9g² +12g - 4

4) 25x² - 30x + 9

5) x² + 6x - 9

6) 49y² + 42y + 36

7) 9m³ + 66m² - 48m

Review: Multiply (x – 2)(x + 2)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

x2 – 4

x -2

x

+2

x2

+2x

-2x

-4

This is called the difference of squares.

x2

+2x-2x-4

Notice the middle terms

eliminate each other!

Difference of Squares

a2 - b2 = (a - b)(a + b)or

a2 - b2 = (a + b)(a - b)

The order does not matter!!

4 Steps for factoringDifference of Squares

1. Are there only 2 terms?2. Is the first term a perfect square?3. Is the last term a perfect square?4. Is there subtraction (difference) in the

problem?If all of these are true, you can factor

using this method!!!

1. Factor x2 - 25When factoring, use your factoring table.

Do you have a GCF?

Are the Difference of Squares steps true?Two terms?

1st term a perfect square?

2nd term a perfect square?

Subtraction?

Write your answer!

No

Yes x2 – 25

Yes

Yes

Yes

( )( )5 xx + 5-

2. Factor 16x2 - 9When factoring, use your factoring table.

Do you have a GCF?

Are the Difference of Squares steps true?Two terms?

1st term a perfect square?

2nd term a perfect square?

Subtraction?

Write your answer!

No

Yes 16x2 – 9

Yes

Yes

Yes

(4x )(4x )3+ 3-

When factoring, use your factoring table.

Do you have a GCF?

Are the Difference of Squares steps true?Two terms?

1st term a perfect square?

2nd term a perfect square?

Subtraction?

Write your answer!(9a )(9a )7b+ 7b-

3. Factor 81a2 – 49b2

No

Yes 81a2 – 49b2

Yes

Yes

Yes

Factor x2 – y2

1. (x + y)(x + y)

2. (x – y)(x + y)

3. (x + y)(x – y)

4. (x – y)(x – y)

Remember, the order doesn’t matter!

When factoring, use your factoring table.

Do you have a GCF?

3(25x2 – 4)

Are the Difference of Squares steps true?Two terms?

1st term a perfect square?

2nd term a perfect square?

Subtraction?

Write your answer! 3(5x )(5x )2+ 2-

4. Factor 75x2 – 12

Yes! GCF = 3

Yes 3(25x2 – 4)

Yes

Yes

Yes

Factor 18c2 + 8d2

1. prime

2. 2(9c2 + 4d2)

3. 2(3c – 2d)(3c + 2d)

4. 2(3c + 2d)(3c + 2d)

You cannot factor using difference of squares because there is no

subtraction!

Factor -64 + 4m2

Rewrite the problem as 4m2 – 64 so the

subtraction is in the middle!

1. prime

2. (2m – 8)(2m + 8)

3. 4(-16 + m2)

4. 4(m – 4)(m + 4)

Ex. 3: Factor completely.

2x² + 18

c² - 5c + 6

5a³ - 80a

8x² - 18x - 35

Ex. 3: Solve each equation.

3x² + 24x + 48 = 049a² + 16 = 56a

z² + 2x + 1= 16 (y – 8)² = 7