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F ACTORING A Q UADRATIC T RINOMIAL

F ACTORING A Q UADRATIC T RINOMIAL. To factor quadratic polynomials whose leading factor is not 1:. b = m q + p n. Find the factors of a ( m and n ). a = mn. Find the factors of c ( p and q ). a x 2 + b x + c = ( m x + p )( n x + q ). The sum of the outer and

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F ACTORING A Q UADRATIC T RINOMIAL

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  1. FACTORING A QUADRATIC TRINOMIAL To factor quadratic polynomials whose leading factor is not 1: b = mq + pn Find the factors of a (m and n). a= mn Find the factors of c (p and q). ax2 + bx + c = (mx + p)(nx + q) The sum of the outer and inner products (mq + pn) is b. c= pq

  2. FACTORING A QUADRATIC TRINOMIAL 22 = 3 •4 + 5 •2 a= mn 6 = 3 • 2 c= pq 6x2 + bx + 20 = (3x + 5)(2x + 4) b = mq + pn c= pq

  3. One Pair of Factors for a and c Factor2x2 + 11x + 5 Test combinations of factors for SOLUTION a 1 and 2 c 1 and 5 Try a= 1 • 2 and c= 1 • 5 (1x + 1) (2x + 5) = 2x2 + 7x + 5 Not correct. Try a= 1 • 2 and c = 5 • 1 (1x + 5) (2x + 1) = 2x2 + 11x + 5 Correct.

  4. A Common Factor for a, b, and c Factor6x2 + 2x – 8 SOLUTION Begin by factoring out the common factor 2. The correct factorization is 6x2 – 2x – 8 = 2(x + 1)(3x – 4). 6x2 + 2x – 8 2 (3x2 – x – 4) = Now factor 3x2 – x – 4 by testing possible factors of a and c. FACTORS OFaANDc PRODUCT CORRECT? When you factor, you can stop testing once you find the correct factorization. a = 1 • 3 and c = (–2)(2) (x – 2)(3x + 2) = 3x2 – 4x – 4 No a = 1 • 3 and c = (2)(–2) (x + 2)(3x – 2) = 3x2 + 4x – 4 No (x – 4)(3x + 1) = 3x2– 11x – 4 a = 1 • 3 and c = (–4)(1) No Yes a = 1 • 3 and c = (1)(–4) (x + 1)(3x – 4) = 3x2 – x – 4

  5. Writing a Quadratic Model 3 2 The solutions are 2 and . – SOLVING QUADRATIC EQUATIONS BY FACTORING A cliff diver jumps from a ledge 48 feet above the ocean with an initial upward velocity of 8 feet per second. How long will it takeuntil the diver enters the water? SOLUTION Use a vertical motion model. Let v = 8 and s = 48. When the diver enters the water, h = 0. Solve the resulting equation for t. h= –16t2+vt+s Vertical motion model 0 = – 16t2 + 8t + 48 Write quadratic model. = – 16t2 + 8t + 48 Substitute values. 0 = (– 8)(2t2 – t – 6) Factor out – 8. 0 = (– 8)(t – 2)(2t + 3) Factor. Negative values do not make sense, so the only reasonable solution is t= 2. It will take the diver 2 seconds.

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