137 1.4 Quadratic Equations Solving for a Specified Variable To solve a quadratic equation for a specified variable, we usually apply the square root property or the quadratic formula. EXAMPLE 8 Solving for a Specified Variable Solve each equation for the specified variable. Use { when taking square roots. (a) = pd 2 4 , for d (b) rt 2 - st = k 1r ≠02, for t SOLUTION (a) = pd 2 4 4 = pd 2 Multiply each side by 4. 4 p = d 2 Divide each side by p. d = {B 4 p Interchange sides; square root property d = {24 2p # 2p 2p Multiply by 1p1p . d = {24 p p Multiply numerators. Multiply denominators. d = {22 p p Simplify the radical. (b) Because rt 2 - st = k has terms with t 2 and t, use the quadratic formula. rt 2 - st - k = 0 Write in standard form. t = -b {2b2 - 4ac 2a Quadratic formula t = -1-s2 {21-s22 - 41r21-k2 21r2 Here a = r, b = -s, and c = -k. t = s {2s2 + 4rk 2r Simplify. S Now Try Exercises 71 and 77. Goal: Isolate d, the specified variable. See the Note following this example. NOTE In Example 8, we took both positive and negative square roots. However, if the variable represents time or length in an application, we consider only the positive square root. The Discriminant The quantity under the radical in the quadratic formula, b2 - 4ac, is the discriminant. x = -b {2b2 - 4ac 2a Discriminant
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