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Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c7-4 Factoring ax2 + bx + c
Holt Algebra 1
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 1
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
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 – 325. z2 + 15z + 366. h2 – 17h + 72
6y2 + 35y + 36
2x2 + 3x – 14
3n2 – 26n + 35
(z + 3)(z + 12) (x – 4)(x + 8)
(h – 8)(h – 9)
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Factor quadratic trinomials of the form ax2 + bx + c.
Objective
Holt McDougal Algebra 1
7-4 Factoring ax2 + 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.
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Example 1: Factoring ax2 + bx + c by Guess and Check
Factor 6x2 + 11x + 4 by guess and check.
( + )( + ) Write two sets of parentheses.
The first term is 6x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
The coefficient of the x2 term is 6. The constant term in the trinomial is 4.
Try factors of 6 for the coefficients and factors of 4 for the constant terms.
(1x + 4)(6x + 1) = 6x2 + 25x + 4 (1x + 2)(6x + 2) = 6x2 + 14x + 4 (1x + 1)(6x + 4) = 6x2 + 10x + 4
(2x + 4)(3x + 1) = 6x2 + 14x + 4
(3x + 4)(2x + 1) = 6x2 + 11x + 4
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Example 1 Continued
Factor 6x2 + 11x + 4 by guess and check.
( + )( + ) Write two sets of parentheses.
The first term is 6x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
6x2 + 11x + 4 = (3x + 4)(2x + 1)
The factors of 6x2 + 11x + 4 are (3x + 4) and (2x + 1).
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 1a
Factor each trinomial by guess and check.
( + )( + ) Write two sets of parentheses.
The first term is 6x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
The coefficient of the x2 term is 6. The constant term in the trinomial is 3.
6x2 + 11x + 3
(1x + 3)(6x + 1) = 6x2 + 19x + 3 (1x + 1)(6x + 3) = 6x2 + 9x + 3
Try factors of 6 for the coefficients and factors of 3 for the constant terms.
(2x + 1)(3x + 3) = 6x2 + 9x + 3
(3x + 1)(2x + 3) = 6x2 + 11x + 3
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 1a Continued
Factor each trinomial by guess and check.
( + )( + ) Write two sets of parentheses.
The first term is 6x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
6x2 + 11x + 3
The factors of 6x2 + 11x + 3 are (3x + 1)(2x + 3).
6x2 + 11x + 3 = (3x + 1)(2x +3)
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 1b Factor each trinomial by guess and check.
( + )( + ) Write two sets of parentheses.
The first term is 3x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
3x2 – 2x – 8
The coefficient of the x2 term is 3. The constant term in the trinomial is –8.
Try factors of 3 for the coefficients and factors of 8 for the constant terms.
(1x – 1)(3x + 8) = 3x2 + 5x – 8
(1x – 8)(3x + 1) = 3x2 – 23x – 8 (1x – 4)(3x + 2) = 3x2 – 10x – 8
(1x – 2)(3x + 4) = 3x2 – 2x – 8
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 1b Continued
Factor each trinomial by guess and check.
( + )( + ) Write two sets of parentheses.
The first term is 3x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
3x2 – 2x – 8
The factors of 3x2 – 2x – 8 are (x – 2)(3x + 4).
3x2 – 2x – 8 = (x – 2)(3x + 4)
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Example 2A: Factoring ax2 + bx + c When c is Positive
Factor each trinomial. Check your answer.
2x2 + 17x + 21
( x + )( x + )a = 2 and c = 21, Outer + Inner = 17.
(x + 7)(2x + 3)
Factors of 2 Factors of 21 Outer + Inner
1 and 2 1 and 21 1(21) + 2(1) = 23
1 and 2 21 and 1 1(1) + 2(21) = 43 1 and 2 3 and 7 1(7) + 2(3) = 13 1 and 2 7 and 3 1(3) + 2(7) = 17
Check (x + 7)(2x + 3) = 2x2 + 3x + 14x + 21= 2x2 + 17x + 21
Use the Foil method.
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 – 16x + 16a = 3 and c = 16, Outer + Inner = –16.
(x – 4)(3x – 4)
Check (x – 4)(3x – 4) = 3x2 – 4x – 12x + 16
= 3x2 – 16x + 16
Use the Foil method.
Factors of 3 Factors of 16 Outer + Inner
1 and 3 –1 and –16 1(–16) + 3(–1) = –19 1 and 3 – 2 and – 8 1( – 8) + 3(–2) = –14 1 and 3 – 4 and – 4 1( – 4) + 3(– 4)= –16
( x + )( x + )
Example 2B: Factoring ax2 + bx + c When c is Positive
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 2a
Factor each trinomial. Check your answer.
6x2 + 17x + 5 a = 6 and c = 5, Outer + Inner = 17.
Factors of 6 Factors of 5 Outer + Inner
1 and 6 1 and 5 1(5) + 6(1) = 11
2 and 3 1 and 5 2(5) + 3(1) = 13 3 and 2 1 and 5 3(5) + 2(1) = 17
(3x + 1)(2x + 5)Check (3x + 1)(2x + 5) = 6x2 + 15x + 2x + 5
= 6x2 + 17x + 5
Use the Foil method.
( x + )( x + )
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 2b
Factor each trinomial. Check your answer.
9x2 – 15x + 4 a = 9 and c = 4, Outer + Inner = –15.
Factors of 9 Factors of 4 Outer + Inner
3 and 3 –1 and – 4 3(–4) + 3(–1) = –15 3 and 3 – 2 and – 2 3(–2) + 3(–2) = –12 3 and 3 – 4 and – 1 3(–1) + 3(– 4)= –15
(3x – 4)(3x – 1)
Check (3x – 4)(3x – 1) = 9x2 – 3x – 12x + 4
= 9x2 – 15x + 4
Use the Foil method.
( x + )( x + )
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
3x2 + 13x + 12 a = 3 and c = 12, Outer + Inner = 13.
Factors of 3 Factors of 12 Outer + Inner
1 and 3 1 and 12 1(12) + 3(1) = 15 1 and 3 2 and 6 1(6) + 3(2) = 12 1 and 3 3 and 4 1(4) + 3(3) = 13
(x + 3)(3x + 4)
Check (x + 3)(3x + 4) = 3x2 + 4x + 9x + 12
= 3x2 + 13x + 12
Use the Foil method.
Check It Out! Example 2c
( x + )( x + )
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Example 3A: Factoring ax2 + bx + c When c is Negative
Factor each trinomial. Check your answer.
3n2 + 11n – 4
( n + )( n+ )a = 3 and c = – 4, Outer + Inner = 11 .
(n + 4)(3n – 1)Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4
= 3n2 + 11n – 4
Use the Foil method.
Factors of 3 Factors of –4 Outer + Inner
1 and 3 –1 and 4 1(4) + 3(–1) = 1 1 and 3 –2 and 2 1(2) + 3(–2) = – 4 1 and 3 –4 and 1 1(1) + 3(–4) = –11
1 and 3 4 and –1 1(–1) + 3(4) = 11
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
2x2 + 9x – 18
( x + )( x+ )a = 2 and c = –18, Outer + Inner = 9.
Factors of 2 Factors of – 18 Outer + Inner
1 and 2 18 and –1 1(– 1) + 2(18) = 35 1 and 2 9 and –2 1(– 2) + 2(9) = 16 1 and 2 6 and –3 1(– 3) + 2(6) = 9
(x + 6)(2x – 3)
Check (x + 6)(2x – 3) = 2x2 – 3x + 12x – 18
= 2x2 + 9x – 18
Use the Foil method.
Example 3B: Factoring ax2 + bx + c When c is Negative
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Factor each trinomial. Check your answer.
4x2 – 15x – 4
( x + )( x+ )a = 4 and c = –4,Outer + Inner = –15.
Factors of 4 Factors of – 4 Outer + Inner
1 and 4 –1 and 4 1(4) + 4(–1) = 0 1 and 4 –4 and 1 1(1) + 4(–4) = –15 1 and 4 –2 and 2 1(2) + 4(–2) = –6
(x – 4)(4x + 1) Use the Foil method.
Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4
= 4x2 – 15x – 4
Example 3C: Factoring ax2 + bx + c When c is Negative
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 3a Factor each trinomial. Check your answer.
6x2 + 7x – 3
( x + )( x+ )a = 6 and c = –3, Outer + Inner = 7.
Factors of 6 Factors of – 3 Outer + Inner
6 and 1 1 and –3 6(–3) + 1(1) = –17 6 and 1 3 and –1 6(–1) + 1(3) = – 3
3 and 2 3(–3) + 2(1) = – 7 3 and 2 3(–1) + 2(3) = 3
1 and –3 3 and –1
2 and 3 2(–3) + 3(1) = – 3 2 and 3 2(–1) + 3(3) = 7
1 and –3 3 and –1
(3x – 1)(2x + 3) Check (3x – 1)(2x + 3) = 6x2 + 9x – 2x – 3
Use the Foil method.
= 6x2 + 7x – 3
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 3b Factor each trinomial. Check your answer.
4n2 – n – 3
( n + )( n+ )
a = 4 and c = –3, Outer + Inner = –1.
(4n + 3)(n – 1) Use the Foil method.
Factors of 4 Factors of –3 Outer + Inner
1 and 4 1 and –3 1(–3) + 4(1) = 1 1 and 4 –1 and 3 1(3) – 4(1) = – 1
Check (4n + 3)(n – 1) = 4n2 – 4n + 3n – 3
= 4n2 – n – 3
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
When the leading coefficient is negative, factor out –1 from each term before using other factoring methods.
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Example 4: Factoring ax2 + bx + c When a is Negative
Factor –2x2 – 5x – 3.
–1(2x2 + 5x + 3)
–1( x + )( x+ )
Factor out –1. a = 2 and c = 3; Outer + Inner = 5
Factors of 2 Factors of 3 Outer + Inner
1 and 2 3 and 1 1(1) + 3(2) = 7 1 and 2 1 and 3 1(3) + 1(2) = 5
–1(x + 1)(2x + 3)
(x + 1)(2x + 3)
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 4a
Factor each trinomial.
–6x2 – 17x – 12
–1(6x2 + 17x + 12)
–1( x + )( x+ )
Factor out –1. a = 6 and c = 12; Outer + Inner = 17
Factors of 6 Factors of 12 Outer + Inner
2 and 3 4 and 3 2(3) + 3(4) = 18 2 and 3 3 and 4 2(4) + 3(3) = 17
(2x + 3)(3x + 4) –1(2x + 3)(3x + 4)
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
Check It Out! Example 4b
Factor each trinomial.
–3x2 – 17x – 10
–1(3x2 + 17x + 10)
–1( x + )( x+ )
Factor out –1. a = 3 and c = 10; Outer + Inner = 17)
Factors of 3 Factors of 10 Outer + Inner
1 and 3 2 and 5 1(5) + 3(2) = 11 1 and 3 5 and 2 1(2) + 3(5) = 17
(3x + 2)(x + 5) –1(3x + 2)(x + 5)
Holt McDougal Algebra 1
7-4 Factoring ax2 + bx + c
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
(–x + 4)(4x + 5)
(3x – 1)(2x – 7)
(2x– 3)(x + 4)
(5x + 2)(x + 3)
(–2x + 1)(x – 3)
(8x + 3)(x + 3)
