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2.6 Solving Quadratic Equations with Complex Roots 11/9/2012 Completing the square Solve x 2 – 2x + 3 = 0 by completing the square. x 2 – 2x + 3 = 0 (x 2 – 2x +1) + 3 – 1 = 0 (x – 1) 2 + 2 = 0 ( x – 1)2 = – 2 Subtract 2 from both sides. x– 1= + – –2 Take the square root of each side. x=1 + – – 2 Add 1 to each side. −2 = −1 • 2 ANSWER = −1 x=1+ – i 2 Write in terms of i. The solutions are 1 + i 2 and 1 – i 2 . Sum of Squares pattern Find complex solution of x2 + 49 = 0 x 2 + 49 = 0 - 49 - 49 x2 = -49 x = -1 • 49 x = ± 7i Sum of Squares pattern Find complex solution of 25x2 + 9 = 0 25x 2 + 9 = 0 -9 -9 25x2 = -9 25 25 9 2 x = 25 x = −1 • x=± 3 i 5 9 25 Quadratic Formula: Is used to solve quadratic equations written in the form ax2 + bx +c = 0 b b 4 ac 2 2a Solve an Equation with Imaginary Solutions –b + b 2 – 4ac – x = 2a Solve x 2 + 2x + 2 = 0. SOLUTION ( –2 + 22 – 4 (1 ( 2 – x = 2 (1 ( ( –4 –2 + – x = 2 2 2i –2 + 2i – x = 2 2 2 x = –1 + – i ANSWER Substitute values in the quadratic formula: a = 1, b = 2, and c = 2. Simplify. Simplify and rewrite using the imaginary unit i. Simplify. The solutions are – 1 + i and – 1 – i. Use the Quadratic Formula Use the quadratic formula to solve the equation. 2 – 4ac + – b b 2 – 2x - x = - 4 x = 2a Rewrite in standard form: 2x2 – x + 4 = 0 −(−1)± (−1)2 −4(2)(4) x= 2(2) 1± −31 x= 4 1± 31 x= 4 1 4 x= ± 31 4 Homework WS 2.6 #1, 2, 4-14even