1. 7-4 Factoring ax2 + bx + c Warm Up Lesson Presentation Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. Warm Up
Find each product.
1. (x – 2)(2x + 7)
Find each trinomial.
2. x2 +4x – 32

3. ESSENTIAL QUESTION
How do you factor quadratic trinomials of the form ax2 + bx + c ?

4. When you multiply (3x + 2)(2x + 5), the coefficient of the x2-term is the product of the coefficients of the x-terms. Also, the constant term in the trinomial is the product of the constants in the binomials.
(3x + 2)(2x + 5) = 6x2 + 19x + 10

5. Example 1: Factoring ax2 + bx + c by Guess and Check
Factor 6x2 + 11x + 4 by guess and check.
( _ + _ )( _ + _ )
Write two sets of parentheses.
( x + _)( _ x + _)
The coefficient of the x2 term is 6. The constant term in the trinomial is 4.
(2x + 4)(3x + 1) = 6x2 + 14x + 4 
(1x + 4)(6x + 1) = 6x2 + 25x + 4 
Try factors of 6 for the coefficients and factors of 4 for the constant terms.
(1x + 2)(6x + 2) = 6x2 + 14x + 4 
(1x + 1)(6x + 4) = 6x2 + 10x + 4 
(3x + 4)(2x + 1) = 6x2 + 11x + 4 

6.     Example 2 Factor each trinomial by guess and check.
6x2 + 11x + 3
( _ + _ )( _ + _ )
Write two sets of parentheses.
( x + _)( _ x + _)
The coefficient of the x2 term is 6. The constant term in the trinomial is 3.
(2x + 1)(3x + 3) = 6x2 + 9x + 3 
Try factors of 6 for the coefficients and factors of 3 for the constant terms.
(1x + 3)(6x + 1) = 6x2 + 19x + 3 
(1x + 1)(6x + 3) = 6x2 + 9x + 3 
(3x + 1)(2x + 3) = 6x2 + 11x + 3 

7.     Example 3 Factor each trinomial by guess and check.
3x2 – 2x – 8
( _ + _ )( _ + _ )
Write two sets of parentheses.
( x + _)( _ x + _)
The coefficient of the x2 term is 3. The constant term in the trinomial is –8.
(1x – 1)(3x + 8) = 3x2 + 5x – 8 
Try factors of 3 for the coefficients and factors of 8 for the constant terms. 
(1x – 4)(3x + 2) = 3x2 – 10x – 8
(1x – 8)(3x + 1) = 3x2 – 23x – 8 
(1x – 2)(3x + 4) = 3x2 – 2x – 8 

8. Example 4: Factoring ax2 + bx + c
When c is negative
3x2 – 16x + 16
( _ + _ )( _ + _ )
a = 3 and c = 16,
Outer + Inner = –16.
( x + _)( _ x + _)
Factors of 3 Factors of 16 Outer + Inner
1 and 3
–1 and –16
1(–16) + 3(–1) = –19
– 2 and – 8
1( – 8) + 3(–2) = –14
– 4 and – 4
1( – 4) + 3(– 4)= –16  
(x – 4)(3x – 4)
Use the Foil method.
Check (x – 4)(3x – 4) = 3x2 – 4x – 12x + 16
= 3x2 – 16x + 16 

9.    Example 5 6x2 + 17x + 5 ( _ + _ )( _ + _ ) a = 6 and c = 5,
( _ + _ )( _ + _ )
a = 6 and c = 5,
Outer + Inner = 17.
( x + _)( _ x + _)
Factors of 6 Factors of 5 Outer + Inner
1 and 6
1 and 5
1(5) + 6(1) = 11
2 and 3
2(5) + 3(1) = 13
3 and 2
3(5) + 2(1) = 17  
(3x + 1)(2x + 5)
Use the Foil method.
Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5
= 6x2 + 17x + 5 

10.    Example 6 9x2 – 15x + 4 ( _ + _ )( _ + _ ) a = 9 and c = 4,
( _ + _ )( _ + _ )
a = 9 and c = 4,
Outer + Inner = –15.
( x + _)( _ x + _)
Factors of 9 Factors of 4 Outer + Inner
3 and 3
–1 and – 4
3(–4) + 3(–1) = –15
– 2 and – 2
3(–2) + 3(–2) = –12
– 4 and – 1
3(–1) + 3(– 4)= –15  
(3x – 4)(3x – 1)
Use the Foil method.
Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4
= 9x2 – 15x + 4 

11.    Example 7 3x2 + 13x + 12 a = 3 and c = 12, Outer + Inner = 13.
( _ + _ )( _ + _ )
( x + _)( _ x + _)
Factors of 3 Factors of 12 Outer + Inner
1 and 3
1 and 12
1(12) + 3(1) = 15
2 and 6
1(6) + 3(2) = 12
3 and 4
1(4) + 3(3) = 13  
(x + 3)(3x + 4)
Use the Foil method.
Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12
= 3x2 + 13x + 12 

12. Example: Factoring ax2 + bx + c
When c is Negative
3n2 + 11n – 4
( _ + _ )( _ + _ )
a = 3 and c = – 4,
Outer + Inner = 11 .
( x + _)( _ x + _)
Factors of 3 Factors of –4 Outer + Inner
1 and 3
–1 and 4
1(4) + 3(–1) = 1
–2 and 2
1(2) + 3(–2) = – 4
–4 and 1
1(1) + 3(–4) = –11 
4 and –1
1(–1) + 3(4) = 11 
(n + 4)(3n – 1)
Use the Foil method.
Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4
= 3n2 + 11n – 4 

13. Example: Factoring ax2 + bx + c
When b & c are Negative
4x2 – 15x – 4
( _ + _ )( _ + _ )
a = 4 and c = –4,
Outer + Inner = –15.
( x + _)( _ x + _)
Factors of 4 Factors of – 4 Outer + Inner
1 and 4
–1 and 4
1(4) + 4(–1) = 0 
1 and 4
–2 and 2
1(2) + 4(–2) = –6 
1 and 4
–4 and 1
1(1) + 4(–4) = –15 
(x – 4)(4x + 1)
Use the Foil method.
Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4
= 4x2 – 15x – 4 

14.   Example 4n2 – n – 3 ( _ + _ )( _ + _ ) a = 4 and c = –3,
( _ + _ )( _ + _ )
a = 4 and c = –3,
Outer + Inner = –1.
( x + _)( _ x + _)
Factors of 4 Factors of –3 Outer + Inner
1 and 4
1 and –3
1(–3) + 4(1) = 1
–1 and 3
1(3) – 4(1) = – 1  
(4n + 3)(n – 1)
Use the Foil method.
Check (4n + 3)(n – 1) = 4n2 – 4n + 3n – 3
= 4n2 – n – 3

15. Example: Factoring ax2 + bx + c
When a is Negative
–2x2 + 7x - 3.
(_ x + )(_ x+ )
Now: a = 2 and c = -3;
Outer + Inner = 5
Factors of -2 Factors of Outer + Inner
1 and -2
3 and -1
-1(1) + 3(-2) = -7  
-3 and 1
1(1) + -2(-3) = 7
(x - 3)(-2x + 1)
ANS: (x - 3)(-2x + 1)

16.   Example –6x2 – 17x – 12 ( _ + _ )( _ + _ ) Factor out –1.
( _ + _ )( _ + _ )
Factor out –1.
-1( x + _)( _ x + _)
a = 6 and c = 12;
Everything positive!
–1(6x2 + 17x + 12)
Factors of 6 Factors of Outer + Inner
2 and 3
4 and 3
2(3) + 3(4) = 18  
3 and 4
2(4) + 3(3) = 17
(2x + 3)(3x + 4)
–1(2x + 3)(3x + 4)

17. Lesson Quiz
Factor each trinomial. Check your answer.
1. 5x2 + 17x + 6
2. 2x2 + 5x – 12
3. 6x2 – 23x + 7
4. –4x2 + 11x + 20
(5x + 2)(x + 3)
(2x– 3)(x + 4)
(3x – 1)(2x – 7)
(–x + 4)(4x + 5)