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)
2. (3y + 4)(2y + 9)
3. (3n – 5)(n – 7)
Factor each trinomial.
4. x2 +4x – 32
5. z2 + 15z + 36
6. h2 – 17h + 72
2x2 + 3x – 14
6y2 + 35y + 36
3n2 – 26n + 35
(x – 4)(x + 8)
(z + 3)(z + 12)
(h – 8)(h – 9)

3. Objective
Factor quadratic trinomials of the form ax2 + bx + c.

4. In the previous lesson you factored trinomials of the form x2 + bx + c
In the previous lesson you factored trinomials of the form x2 + bx + c. Now you will factor trinomials of the form ax2 + bx + c, where a ≠ 0.

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 first term is 6x2, so at least one variable term has a coefficient other than 1.
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. Factor 6x2 + 11x + 4 by guess and check.
Example 1 Continued
Factor 6x2 + 11x + 4 by guess and check.
( )( )
Write two sets of parentheses.
( x + )( x + )
The first term is 6x2, so at least one variable term has a coefficient other than 1.
The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1).
6x2 + 11x + 4 = (3x + 4)(2x + 1)

7.     Check It Out! Example 1a
Factor each trinomial by guess and check.
6x2 + 11x + 3
( )( )
Write two sets of parentheses.
( x + )( x + )
The first term is 6x2, so at least one variable term has a coefficient other than 1.
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 

8. Check It Out! Example 1a Continued
Factor each trinomial by guess and check.
6x2 + 11x + 3
( )( )
Write two sets of parentheses.
( x + )( x + )
The first term is 6x2, so at least one variable term has a coefficient other than 1.
The factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3).
6x2 + 11x + 3 = (3x + 1)(2x +3)

9.     Check It Out! Example 1b
Factor each trinomial by guess and check.
3x2 – 2x – 8
( )( )
Write two sets of parentheses.
The first term is 3x2, so at least one variable term has a coefficient other than 1.
( 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 

10. Check It Out! Example 1b Continued
Factor each trinomial by guess and check.
3x2 – 2x – 8
( )( )
Write two sets of parentheses.
( x + )( x + )
The first term is 3x2, so at least one variable term has a coefficient other than 1.
The factors of 3x2 – 2x – 8 are (x – 2)(3x + 4).
3x2 – 2x – 8 = (x – 2)(3x + 4)

11. Example 2A: Factoring ax2 + bx + c When c is Positive
Factor each trinomial. Check your answer.
2x2 + 17x + 21
a = 2 and c = 21, Outer + Inner = 17.
( x + )( x + )
Factors of 2 Factors of 21 Outer + Inner
1 and 2
1 and 21
1(21) + 2(1) = 23
21 and 1
1(1) + 2(21) = 43
3 and 7
1(7) + 2(3) = 13
7 and 3
1(3) + 2(7) = 17  
(x + 7)(2x + 3)
Use the Foil method.
Check (x + 7)(2x + 3) = 2x2 + 3x + 14x + 21
= 2x2 + 17x + 21 

12. Example 2B: Factoring ax2 + bx + c When c is Positive
Factor each trinomial. Check your answer.
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 

13.    Check It Out! Example 2a
Factor each trinomial. Check your answer.
6x2 + 17x + 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 

14.    Check It Out! Example 2b
Factor each trinomial. Check your answer.
9x2 – 15x + 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 

15.    Check It Out! Example 2c
Factor each trinomial. Check your answer.
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 

16. Example 3A: Factoring ax2 + bx + c When c is Negative
Factor each trinomial. Check your answer.
3n2 + 11n – 4
a = 3 and c = – 4,
Outer + Inner = 11 .
( n + )( n+ )
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 

17. Example 3B: Factoring ax2 + bx + c When c is Negative
Factor each trinomial. Check your answer.
2x2 + 9x – 18
a = 2 and c = –18,
Outer + Inner = 9.
( x + )( x+ )
Factors of 2 Factors of – 18 Outer + Inner
1 and 2
18 and –1
1(– 1) + 2(18) = 35
9 and –2
1(– 2) + 2(9) = 16
6 and –3
1(– 3) + 2(6) = 9  
(x + 6)(2x – 3)
Use the Foil method.
Check (x + 6)(2x – 3) = 2x2 – 3x + 12x – 18
= 2x2 + 9x – 18 

18. Example 3C: Factoring ax2 + bx + c When c is Negative
Factor each trinomial. Check your answer.
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 

19.   Check It Out! Example 3a Factor each trinomial. Check your answer.
6x2 + 7x – 3
a = 6 and c = –3,
Outer + Inner = 7.
( x + )( x+ )
Factors of 6 Factors of – 3 Outer + Inner
6 and 1
1 and –3
6(–3) + 1(1) = –17
3 and –1
6(–1) + 1(3) = – 3  
3 and 2
3(–3) + 2(1) = – 7
3(–1) + 2(3) = 3
2 and 3
2(–3) + 3(1) = – 3
2(–1) + 3(3) = 7
(3x – 1)(2x + 3)
Use the Foil method.
Check (3x – 1)(2x + 3) = 6x2 + 9x – 2x – 3
= 6x2 + 7x – 3

20.   Check It Out! Example 3b Factor each trinomial. Check your answer.
4n2 – n – 3
a = 4 and c = –3,
Outer + Inner = –1.
( n + )( n+ )
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

21. When the leading coefficient is negative, factor out –1 from each term before using other factoring methods.

22. Example 4: Factoring ax2 + bx + c When a is Negative
Factor –2x2 – 5x – 3.
–1(2x2 + 5x + 3)
Factor out –1.
a = 2 and c = 3;
Outer + Inner = 5
–1( x + )( x+ )
Factors of 2 Factors of Outer + Inner
1 and 2
3 and 1
1(1) + 3(2) = 7  
1 and 3
1(3) + 1(2) = 5
(x + 1)(2x + 3)
–1(x + 1)(2x + 3)

23.   Check It Out! Example 4a Factor each trinomial. –6x2 – 17x – 12
Factor out –1.
–1(6x2 + 17x + 12)
a = 6 and c = 12;
Outer + Inner = 17
–1( x + )( x+ )
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)

24.   Check It Out! Example 4b Factor each trinomial. –3x2 – 17x – 10
Factor out –1.
–1(3x2 + 17x + 10)
a = 3 and c = 10;
Outer + Inner = 17)
–1( x + )( x+ )
Factors of 3 Factors of Outer + Inner
1 and 3
2 and 5
1(5) + 3(2) = 11  
5 and 2
1(2) + 3(5) = 17
(3x + 2)(x + 5)
–1(3x + 2)(x + 5)

25. Lesson Quiz
Factor each trinomial.
1. 5x2 + 17x + 6
2. 2x2 + 5x – 12
3. 6x2 – 23x + 7
4. –4x2 + 11x + 20
5. –2x2 + 7x – 3
6. 8x2 + 27x + 9
(5x + 2)(x + 3)
(2x– 3)(x + 4)
(3x – 1)(2x – 7)
(–x + 4)(4x + 5)
(–2x + 1)(x – 3)
(8x + 3)(x + 3)