Equations in various forms may be solved using the quadratic formula. EXAMPLE 2 Solve using the quadratic formula. (3p 1)(3p 4) 3p 5 Solution Is (3p 1)(3p 4) 3p 5 in the form ax2 bx c 0? No. Before we can apply the quadratic formula, we must write it in that form. (3p 1)(3p 4) 3p 5 9p2 9p 4 3p 5 Multiply using FOIL. 9p2 6p 1 0 Subtract 3p, and add 5 to both sides. The equation is in the correct form. Identify a, b, and c: a 9 b 6 c 1 p b 2b2 4ac 2a (6) 2(6)2 4(9)(1) 2(9) 6 136 36 18 6 10 18 6 0 18 6 18 The solution set is e 1 3 f . YOU TRY 2 Solve using the quadratic formula. Quadratic formula Substitute a 9, b 6, and c 1. Perform the operations. 1 3 a) 3 2z 2z2 b) (d 6)(d 2) 10 To solve a quadratic equation containing fractions, fi rst multiply by the LCD to eliminate the fractions. Then, solve using the quadratic formula. 3 Determine the Number and Type of Solutions of a Quadratic Equation Using the Discriminant We can fi nd the solutions of any quadratic equation of the form ax2 bx c 0 (a 0) using the quadratic formula. x b 2b2 4ac 2a The radicand in the quadratic formula determines the type of solution a quadratic equation has. www.mhhe.com/messersmith SECTION 10.2 The Quadratic Formula 627
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