Step 4: Factor. ax 3 4 b 2 17 16 ax c c 3 4 is 1 2 a3 2 b, the coeffi cient of x. Step 5: Solve using the square root property. ax 3 4 b 2 17 16 ax b 2a b b 2a is b 2a 2 b 1 2 ab a 2 b2 4ac 4a2 b, the coeffi cient of x. b2 4ac 4a2 x 3 4 A 17 16 x b 2a B b2 4ac 4a2 x 3 4 117 4 116 4 x b 2a 2b2 4ac 2a 24a2 2a x 3 4 117 4 Subtract 3 4 . x b 2a 2b2 4ac 2a Subtract b 2a . x 3 117 4 Same denominators, x b 2b2 4ac 2a add numerators. The result on the right is called the quadratic formula. Same denominators, add numerators. Definition 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. Note 1) To use the quadratic formula, write the equation to be solved in the form ax2 bx c 0 so that a, b, and c can be identified correctly. 2) x b 2b2 4ac 2a represents the two solutions x b 2b2 4ac 2a and x b 2b2 4ac 2a . 3) Notice that the fraction bar runs under b and under the radical. x b 2b2 4ac 2a x b 2b2 4ac 2a Correct Incorrect 4) When deriving the quadratic formula, using the allows us to say that 24a2 2a. 5) The quadratic formula is a very important result, and we will use it often. It should be memorized! Memorize this formula! www.mhhe.com/messersmith SECTION 10.2 The Quadratic Formula 625
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