Definition/Procedure Example 10.2 The Quadratic Formula The Quadratic Formula The solutions of any quadratic equation of the form ax2 bx c 0 (a 0) are x b 2b2 4ac 2a This formula is called the quadratic formula. (p. 625) The expression under the radical, b2 4ac is called the discriminant. 1) If b2 4ac is positive and the square of an integer, the equation has two rational solutions. 2) If b2 4ac is positive but not a perfect square, the equation has two irrational solutions. 3) If b2 4ac is negative, the equation has two nonreal, complex solutions of the form a bi and a bi. 4) If b2 4ac 0, the equation has one rational solution. (p. 628) Solve 2x2 5x 2 0 using the quadratic formula. a 2 b 5 c 2 Substitute the values into the quadratic formula, and simplify. x (5) 2(5)2 4(2)(2) 2(2) x 5 125 16 4 5 141 4 The solution set is e 5 241 4 , 5 141 4 f . Find the value of the discriminant for 3m2 4m 5 0, and determine the number and type of solutions of the equation. a 3 b 4 c 5 b2 4ac (4)2 4(3)(5) 16 60 44 Discriminant 44. The equation has two nonreal, complex solutions of the form a bi and a bi. 10.3 Equations in Quadratic Form Some equations that are not quadratic can be solved using the same methods that can be used to solve quadratic equations. These are called equations in quadratic form. (p. 638) 10.4 Formulas and Applications Solve a Formula for a Variable. (p. 645) Solve r4 2r2 24 0. (r2 4)(r2 6) 0 Factor. b R r2 4 0 or r2 6 0 r2 4 r2 6 r 14 r 16 r 2 r i16 The solution set is {i16, i16, 2, 2}. Solve for s: g 10 s2 s2g 10 Multiply both sides by s2. s2 10 g Divide both sides by g. s A 10 g Square root property s 110 1g 1g 1g Rationalize the denominator. s 210g g www.mhhe.com/messersmith CHAPTER 10 Summary 693
messersmith_power_intermediate_algebra_1e_ch4_7_10
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