A64 APPENDIX Review Because we have defined the square root of a negative number, we now restate the quadratic formula without restriction. Solving a Quadratic Equation in the Complex Number System Solve the equation − + = x x4 8 0 2 in the complex number system. Solution EXAMPLE 11 Here = = − = a b c 1, 4, 8, and ( ) − = − − ⋅ ⋅ = − b ac 4 4 4 1 8 16. 2 2 Using equation (9), we find that x i i i i 4 16 2 1 4 16 2 4 4 2 2 2 2 2 2 2 ( ) ( ) = − − ± − ⋅ = ± = ± = ± = ± The equation has the solution set { } − + i i 2 2 , 2 2 . Check: i i i i i i i i i i i i i 2 2 : 2 2 422 848 4 88 8 4 4 4 4 0 2 2 : 2 2 422 848 4 88 8 4 4 0 2 2 2 2 2 ( ) ( ) ( ) ( ) + + − ++=++−−+ = + = − = − − − −+=−+−++ = − = Character of the Solutions of a Quadratic Equation In the complex number system, consider a quadratic equation + + = ax bx c 0, 2 where a 0 ≠ and a b , , and c are real numbers. • If − > b ac 4 0, 2 the equation has two unequal real solutions. • If − = b ac 4 0, 2 the equation has a repeated real solution, a double root. • If − < b ac 4 0, 2 the equation has two complex solutions that are not real. The solutions are conjugates of each other. THEOREM Quadratic Formula In the complex number system, the solutions of the quadratic equation + + = ax bx c 0, 2 where a , b , and c are real numbers and ≠ a 0, are given by the formula = − ± − x b b ac a 4 2 2 (9) Figure 26 shows the check of the solution using a TI-84 Plus CE graphing calculator. The discriminant − b ac 4 2 of a quadratic equation still serves as a way to determine the character of the solutions. The third conclusion in the display is a consequence of the fact that if − = − < b ac N 4 0, 2 then by the quadratic formula, the solutions are = − + − = − + − = − + = − + x b b ac a b N a b Ni a b a N a i 4 2 2 2 2 2 2 and = − − − = − − − = − − = − − x b b ac a b N a b Ni a b a N a i 4 2 2 2 2 2 2 which are conjugates of each other. Now Work problem 65 Figure 26
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