136 CHAPTER 1 Equations and Inequalities Throughout this text, unless otherwise specified, we use the set of complex numbers as the domain when solving equations of degree 2 or greater. CAUTION Remember to extend the fraction bar in the quadratic formula under the −b term in the numerator. EXAMPLE 6 Using the Quadratic Formula (Nonreal Complex Solutions) Solve 2x2 = x - 4. SOLUTION 2x2 - x + 4 = 0 Write in standard form. x = -1-12 {21-122 - 4122142 2122 Quadratic formula with a = 2, b = -1, c = 4 x = 1 {21 - 32 4 x = 1 {2-31 4 Simplify. x = 1 {i 231 4 2-1 = i The solution set is E1 4 { 231 4 i F . S Now Try Exercise 57. Use parentheses and substitute carefully to avoid errors. The equation x3 + 8 = 0 is a cubic equation because the greatest degree of the terms is 3. While a quadratic equation (degree 2) can have as many as two solutions, a cubic equation (degree 3) can have as many as three solutions. The maximum possible number of solutions corresponds to the degree of the equation. EXAMPLE 7 Solving a Cubic Equation Solve x3 + 8 = 0 using factoring and the quadratic formula. SOLUTION x3 + 8 = 0 1x + 221x2 - 2x + 42 = 0 Factor as a sum of cubes. x + 2 = 0 or x2 - 2x + 4 = 0 Zero-factor property x = -2 or x = -1-22 {21-222 - 4112142 2112 Quadratic formula with a = 1, b = -2, c = 4 x = 2{2-12 2 Simplify. x = 2{2i 23 2 Simplify the radical. x = 2 A 1{i 23 B 2 Factor out 2 in the numerator. x = 1{i 23 Divide out the common factor. The solution set is E -2, 1{i 23F. S Now Try Exercise 67.
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