47) 6k(k 4) 3 5(k2 12) 8k {9, 7} 48) 7b(b 1) 15 6(b2 2) 11b {1, 3} 49) 3(n2 15) 4n 4n(n 3) 19 {8} 50) 8( p2 6) 9p 3p(3p 7) 13 {7, 5} Objective 3: Solve Higher-Degree Equations by Factoring 51) To solve 5t3 20t 0, Julio begins by dividing the equation by 5t to get t2 4 0. Is this correct? Why or why not? No. You cannot divide an equation by a variable because you may eliminate a solution and may be dividing by zero. 52) What are two possible fi rst steps for solving 5t3 20t 0? i) Divide by 5 to get t3 4t 0, then factor out t to get t(t2 4) 0; or ii) Factor out 5t to get 5t(t2 4) 0. Determine whether the statements in Exercises 55 and 56 are always, sometimes, or never true. 53) The equation ax2 bx c 0 has two different solutions. sometimes 54) The solutions of x2 bx c 0 are opposites of the factors of c that have b as their sum. always The following equations are not quadratic but can be solved by factoring and applying the zero product rule. Solve each equation. 55) 7w(8w 9)(w 9 6) 0 e6, 0, f 8 56) 5q(4q 7)(q 7 3) 0 e 0, , 3 f 4 57) (6m 7)(m2 5m 6) 0 e 7 6 , 2, 3 f 2 9 58) (9c 2)(c2 9c 8) 0 e8, 1, f 59) 49h h3 {0, 7, 7} 60) r3 36r {0, 6, 6} 61) 5w2 36w w3 62) 10p2 25p p3 {4, 0, 9} 63) 60a 44a2 8a3 5 e 0, , 3 f 64) 6z3 16z 50z2 2 65) 162b3 8b 0 66) 75x 27x3 e8, e 0, 2 9 , 2 9 f e 0, 5 3 , 5 3 f Mixed Exercises: Objectives 1–3 Solve each equation. 67) 63 4y(y 8) 9 f 68) 84 g(g 19) 2 69) 6d 1 9d2 70) (9p 2)(p2 10p 11) 0 71) a2 a 30 {5, 6} 72) 45k 27 18k2 73) 48t 3t3 {0, 4, 4} 74) (c 8)(c 6) 1 {7} 75) 104r 36 12r2 e 1 3 , 9 f e 7 2 , e 1 3 f e1, 76) 3t(t 5) 14 5 t(t 3) e 3 2 f 77) w2 121 0 {11, 11} 78) h2 15h 54 0 {9, 6} 79) (2n 5)(n2 6n 9) 0 e 5 3 , 3 f 80) 36b2 60b 0 e 5 3 , 0 f {0, 5} 1 3 , 0 f {12, 7} 2 9 , 11 f e 1 2 , 3 f R3) Do you think graphs of quadratic equations will look similar based on the number of solutions? R1) Good factoring skills will help you solve quadratic equations quickly. Which techniques could you improve? R2) Explain how the sequence of the objectives in this chapter has helped you develop the skills needed to solve quadratic equations. www.mhhe.com/messersmith SECTION 7.5 Solving Quadratic Equations by Factoring 431
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