190 CHAPTER 1 Equations and Inequalities Concepts Examples Divide complex numbers by multiplying the numerator and denominator by the complex conjugate of the denominator. Solve. 6x2 + x - 1 = 0 13x - 1212x + 12 = 0 Factor. 3x - 1 = 0 or 2x + 1 = 0 Zero-factor property x = 1 3 or x = - 1 2 Solve each equation. The solution set is E - 1 2 , 1 3F. x2 = 12 x = {212 = {223 The solution set is E {223F. x2 + 2x + 3 = 0 x = -2{222 - 4112132 2112 Quadratic formula with a = 1, b = 2, c = 3 x = -2{2-8 2 Simplify. x = -2{2i22 2 2-8 = 2-4 # 2 = 2i22 x = 2 A -1{i22 B 2 Factor out 2 in the numerator. x = -1{i22 Divide out the common factor. The solution set is E -1{i22 F . 1.4 Quadratic Equations Zero-Factor Property If a and b are complex numbers with ab = 0, then a = 0 or b = 0 or both equal 0. Square Root Property The solution set of x2 = k is E!k, −!k F , abbreviated E t!k F . Quadratic Formula The solutions of the quadratic equation ax2 + bx + c = 0, where a≠0, are given by the quadratic formula. x = −b t!b 2 −4 ac 2 a 3 + i 1 + i = 13 + i211 - i2 11 + i211 - i2 Multiply by 1 - i 1 - i . = 3 - 3i + i - i2 1 - i2 Multiply. = 4 - 2i 2 Combine like terms; i2 = -1 = 4 2 - 2i 2 a + bi c = a c + bi c = 2 - i Write in lowest terms and standard form.
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