1. 4.5 Supplemental Notes - Factoring Special Products - Solutions
Algebra 3
4.5 Supplemental Notes - Factoring Special Products - Solutions

2. Use the Patterns! (2x + 3)2 =4x² + 12x + 9 =4p² - 16 (2p - 4) (2p + 4)
First and last terms are perfect squares!
(2x + 3)2
=4x² + 12x + 9
Perfect Square Trinomial!
The middle term is twice the product of the square roots
of the first and third terms.
=4p² - 16
(2p - 4) (2p + 4)
Difference of two squares (DTS)!
The difference of…
two squares!
First and last terms are perfect squares!
(2x - y)2
=4x² - 4xy + y²
Perfect Square Trinomial!
The middle term is twice the product of the square roots
of the first and third terms.
The key is to recognize when you see a perfect square trinomial
or a difference of squares!

4. Examples:
1) 𝑥 2 +10𝑥+25 a=x, b=5
= (𝑥) 𝑥 + (5) 2
=(𝑥+5)(𝑥+5)
= (𝑥+5) 2

5. Examples:
2) 9𝑥 2 +42𝑥+49 a=3x, b=7
= (3𝑥) 𝑥 + (7) 2
=(3𝑥+7)(3𝑥+7)
= (3𝑥+7) 2

6. Examples:
3) 𝑥 2 −10𝑥+25 a=x, b=5
= (𝑥) 2 −2 5 𝑥
=(𝑥−5)(𝑥−5)
= (𝑥−5) 2

7. Examples:
4) 100𝑥 2 −60𝑥+9 a=10x, b=3
= (10𝑥) 2 − 𝑥
=(10𝑥−3)(10𝑥−3)
= (10𝑥−3) 2

8. Once you can see the patterns easily, you can factor in one step:
𝑥 2 +10𝑥+25 a=x, b=5
= (𝑥+5) 2
9𝑥 2 +42𝑥+49 a=3x, b=7
= (3𝑥+7) 2
𝑥 2 −10𝑥+25 a=x, b=5
= (𝑥−5) 2
100𝑥 2 −60𝑥+9 a=10x, b=3
= (10𝑥−3) 2

9. Factoring Patterns! a² + 2ab + b2 =(a + b)2 a² - b2 =(a - b)(a + b)
First and last terms are perfect squares!
a² + 2ab + b2
=(a + b)2
Perfect Square Trinomial!
The middle term is twice the product of the square roots
of the first and third terms.
a² - b2
=(a - b)(a + b)
Difference of two squares (DTS)!
The difference of…
two squares!
First and last terms are perfect squares!
a² - 2ab + b²
=(a - b)2
Perfect Square Trinomial!
The middle term is twice the product of the square roots
of the first and third terms.
The key is to recognize when you see a perfect square trinomial
or a difference of squares!

10. Factor! 2x²- 18 =2(x + 3)(x – 3) =2(x²- 9) =(7t + ½r)(7t – ½r)
Difference of Squares!
=(7t + ½r)(7t – ½r)
49t²- ¼r2
Difference of Squares!
81x²- 25y²
=(9x – 5y)(9x + 5y)
Difference of Squares!
27x²- 12
=3(3x + 2)(3x – 2)
=3(9x²- 4)
Difference of Squares!

11. Factor! -3x²- 18x - 27 =-3(x + 3)2 =-3(x²+ 6x + 9) =(3y – 10)2
Perfect Square Trinomial!
=(3y – 10)2
9y²- 60y + 100
Perfect Square Trinomial!
2x²- 12x + 18
=2(x – 3)2
=2(x²- 6x + 9)
Perfect Square Trinomial!
49x²+ 84x + 36
=(7x + 6)2
Perfect Square Trinomial!

12. Solve! (x – 5)2 = 0 (x-5)(x-5)=0 3x²- 30x = -75 x-5=0 or x-5=0
Divide each side by 3!
x²- 10x + 25 = 0
x = 5
Perfect Square Trinomial!
(6y + 11)(6y – 11) = 0
6y+11=0 6y-11=0
6y=-11 6y=11
Y=-11/6 y=11/6
36y²- 121 = 0
Difference of Squares!
y = {-11/6, 11/6}

13. Solve! -6x²+ 8x + 14 = 0 (x )(3x ) = 0 x+1=0 3x-7=0 x=-1 3x=7 x=7/3
Divide each side by -2!
3x²- 4x – 7 = 0
x = {-1, 7/3}

14. Solve! 4x²- 1 = 0 (2x + 1)(2x – 1) = 0 2x+1=0 2x-1=0 2x=-1 2x=1
Difference of Squares!
7x²- 10x = -3
7x²- 10x + 3 = 0
x = {-½, ½}
(7x )(x ) = 0
7x-3=0 x-1=0
7x=3 x=1
x=3/7
– –
x = {3/7, 1}

15. Solve! 32x²- 80x + 50 = 0 (4x – 5)2 = 0 4x-5=0 4x=5 16x²- 40x + 25 = 0
Divide each side by 2!
16x²- 40x + 25 = 0
Perfect Square Trinomial!
x = 5/4

16. HW
Find and try at least one problem that fits each of the special product factoring rules:
(look in 4.4 and 4.5)
a² + 2ab + b2
=(a + b)2
a² - b2
=(a - b)(a + b)
a² - 2ab + b²
=(a - b)2