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Section 4: The Quadratic Formula

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 5.4 The Quadratic Formula 481 Version: Fall 2007 5.4 The Quadratic Formula Consider the general quadratic function f ( x ) = ax 2 + bx + c. In the previous section, we learned that we can find the zeros of this function by solving the equation f ( x ) = 0 . If we substitute f ( x ) = ax 2 + bx + c , then the resulting equation ax 2 + bx + c = 0 (1) is called a quadratic equation . In the previous section, we solved equations of this type by factoring and using the zero product property. However, it is not always possible to factor the trinomial on the left-hand side of the quadratic equation (1) as a product of factors with integer coefficients. For example, consider the quadratic equation 2 x 2 + 7 x 3 = 0 . (2) Comparing 2 x 2 + 7 x 3 with ax 2 + bx + c , let’s list all integer pairs whose product is ac = (2)( 3) = 6 . 1 , 6 1 , 6 2 , 3 2 , 3 Not a single one of these integer pairs adds to b = 7 . Therefore, the quadratic trinomial 2 x 2 + 7 x 3 does not factor over the integers. 2 Consequently, we’ll need another method to solve the quadratic equation (2) . The purpose of this section is to develop a formula that will consistently provide solutions of the general quadratic equation (1) . However, before we can develop the “Quadratic Formula,” we need to lay some groundwork involving the square roots of numbers. Square Roots We begin our discussion of square roots by investigating the solutions of the equation x 2 = a . Consider the rather simple equation x 2 = 25 . (3) Because ( 5) 2 = 25 and (5) 2 = 25 , equation (3) has two solutions, x = 5 or x = 5 . We usually denote these solutions simultaneously, using a “plus or minus” sign: Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 This means that the trinomial 2 x 2 + 7 x 3 cannot be expressed as a product of factors with integral 2 (integer) coefficients.
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482 Chapter 5 Quadratic Functions Version: Fall 2007 x = ± 5 These solutions are called square roots of 25. Because there are two solutions, we need a different notation for each. We will denote the positive square root of 25 with the notation 25 and the negative square root of 25 with the notation 25 . Thus, 25 = 5 and 25 = 5 . In a similar vein, the equation x 2 = 36 has two solutions, x = ± 36 , or alternatively, x = ± 6 . The notation 36 calls for the positive square root, while the notation 36 calls for the negative square root. That is, 36 = 6 and 36 = 6 . It is not necessary that the right-hand side of the equation x 2 = a be a “perfect square.” For example, the equation x 2 = 7 has solutions x = ± 7 . (4) There is no rational square root of 7. That is, there is no way to express the square root of 7 in the form p/q , where p and q are integers. Therefore, 7 is an example of an irrational number. However, 7 is a perfectly valid real number and we’re perfectly comfortable leaving our answer in the form shown in equation (4) .
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