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

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

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

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

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 

    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 

    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 

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 

   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 

   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 

   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 

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 

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 

  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

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 -3 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)

  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 12 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)

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)