161 1.6 OtherTypes of Equations and Applications CHECK Let x = 243 in the original equation. x3/5 = 2433/5 = A 25 243 B 3 = 33 = 27 ✓ True The solution set is 52436. (b) 1x - 422/3 = 16 C 1x - 422/3D 3/2 = {163/2 x - 4 = {64 {163/2 = {A 116 B 3 = {43 = {64 x = 4{64 Add 4 to each side. x = -60 or x = 68 Proposed solutions CHECK 1x - 422/3 = 16 Original equation Raise each side to the power 3 2. Insert { because this involves an even root, as indicated by the 2 in the denominator. 1-60 - 422/3 ≟16 Let x = -60. 1-6422/3 ≟16 A 2 3 -64 B 2 ≟16 16 = 16 ✓ True 168 - 422/3 ≟16 Let x = 68. 642/3 ≟16 A 2 3 64 B 2 ≟16 16 = 16 ✓ True Both proposed solutions check, so the solution set is 5-60, 686. S Now Try Exercises 75 and 79. Equations Quadratic in Form Many equations that are not quadratic equations can be solved using similar methods. The equation 1x + 122/3 - 1x + 121/3 - 2 = 0 is not a quadratic equation in x. However, with the substitutions u = 1x + 121/3 and u2 = 31x + 121/342 = 1x + 122/3, the equation becomes u2 - u - 2 = 0, which is a quadratic equation in u. This quadratic equation can be solved to find u, and then u = 1x + 121/3 can be used to find the values of x, the solutions to the original equation. Equation Quadratic in Form An equation is quadratic in form if it can be written as au2 +bu +c =0, where a≠0 and u is some algebraic expression.
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