SECTION A.7 Complex Numbers; Quadratic Equations in the Complex Number System A63 Now Work problems 35 and 43 NOTE Another way to find i ( ) +2 3 is to multiply out i i ( ) ( ) + + 2 2 . 2 j Evaluating the Square Root of a Negative Number (a) i i 1 1 − = = (b) i i 16 16 4 − = = (c) i i 8 8 2 2 − = = EXAMPLE 9 Using this special product formula, ( ) ( ) ( ) + = + ⋅ ⋅ + ⋅ ⋅ + = + + − + − = + i i i i i i i 2 2 3 2 3 2 8 12 6 1 2 11 3 3 2 2 3 2 Solve Quadratic Equations in the Complex Number System Quadratic equations with a negative discriminant have no real number solution. However, if we extend our number system to allow complex numbers, quadratic equations always have solutions. Since the solution to a quadratic equation involves the square root of the discriminant, we begin with a discussion of square roots of negative numbers. CAUTION In writing i − =N N , be sure to place i outside the symbol. j DEFINITION Principal Square Root of N− If N is a positive real number, we define the principal square root of N− , denoted by −N, as N Ni − = where i is the imaginary unit and = − i 1. 2 Now Work problem 51 Solving Equations Solve each equation in the complex number system. (a) = x 4 2 (b) = − x 9 2 Solution EXAMPLE 10 (a) = = ± = ± x x 4 4 2 2 The equation has two solutions, −2 and 2. The solution set is { } −2, 2 . (b) x x i i 9 9 9 3 2 = − = ± − = ± = ± The equation has two solutions, − i3 and i3 . The solution set is { } − i i 3 , 3 . CAUTION When working with square roots of negative numbers, do not set the square root of a product equal to the product of the square roots (which can be done with positive real numbers). To see why, look at this calculation: We know that = 100 10. However, it is also true that ( ) ( ) = − − 100 25 4 , so i i i i i ( ) ( ) = = − − = − − = ⋅ = ⋅ = =− 10 100 25 4 254 25 4 52 10 10 2 ↑ Here is the error. j Now Work problem 59
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