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Simplifying (8 + 9i)(1 + -4i) + -1(8 + 7i)(8 + -7i) = 0 Multiply (8 + 9i) * (1 + -4i) (8(1 + -4i) + 9i * (1 + -4i)) + -1(8 + 7i)(8 + -7i) = 0 ((1 * 8 + -4i * 8) + 9i * (1 + -4i)) + -1(8 + 7i)(8 + -7i) = 0 ((8 + -32i) + 9i * (1 + -4i)) + -1(8 + 7i)(8 + -7i) = 0 (8 + -32i + (1 * 9i + -4i * 9i)) + -1(8 + 7i)(8 + -7i) = 0 (8 + -32i + (9i + -36i2)) + -1(8 + 7i)(8 + -7i) = 0 Combine like terms: -32i + 9i = -23i (8 + -23i + -36i2) + -1(8 + 7i)(8 + -7i) = 0 Multiply (8 + 7i) * (8 + -7i) 8 + -23i + -36i2 + -1(8(8 + -7i) + 7i * (8 + -7i)) = 0 8 + -23i + -36i2 + -1((8 * 8 + -7i * 8) + 7i * (8 + -7i)) = 0 8 + -23i + -36i2 + -1((64 + -56i) + 7i * (8 + -7i)) = 0 8 + -23i + -36i2 + -1(64 + -56i + (8 * 7i + -7i * 7i)) = 0 8 + -23i + -36i2 + -1(64 + -56i + (56i + -49i2)) = 0 Combine like terms: -56i + 56i = 0 8 + -23i + -36i2 + -1(64 + 0 + -49i2) = 0 8 + -23i + -36i2 + -1(64 + -49i2) = 0 8 + -23i + -36i2 + (64 * -1 + -49i2 * -1) = 0 8 + -23i + -36i2 + (-64 + 49i2) = 0 Reorder the terms: 8 + -64 + -23i + -36i2 + 49i2 = 0 Combine like terms: 8 + -64 = -56 -56 + -23i + -36i2 + 49i2 = 0 Combine like terms: -36i2 + 49i2 = 13i2 -56 + -23i + 13i2 = 0 Solving -56 + -23i + 13i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -4.307692308 + -1.769230769i + i2 = 0 Move the constant term to the right: Add '4.307692308' to each side of the equation. -4.307692308 + -1.769230769i + 4.307692308 + i2 = 0 + 4.307692308 Reorder the terms: -4.307692308 + 4.307692308 + -1.769230769i + i2 = 0 + 4.307692308 Combine like terms: -4.307692308 + 4.307692308 = 0.000000000 0.000000000 + -1.769230769i + i2 = 0 + 4.307692308 -1.769230769i + i2 = 0 + 4.307692308 Combine like terms: 0 + 4.307692308 = 4.307692308 -1.769230769i + i2 = 4.307692308 The i term is -1.769230769i. Take half its coefficient (-0.8846153845). Square it (0.7825443785) and add it to both sides. Add '0.7825443785' to each side of the equation. -1.769230769i + 0.7825443785 + i2 = 4.307692308 + 0.7825443785 Reorder the terms: 0.7825443785 + -1.769230769i + i2 = 4.307692308 + 0.7825443785 Combine like terms: 4.307692308 + 0.7825443785 = 5.0902366865 0.7825443785 + -1.769230769i + i2 = 5.0902366865 Factor a perfect square on the left side: (i + -0.8846153845)(i + -0.8846153845) = 5.0902366865 Calculate the square root of the right side: 2.256155289 Break this problem into two subproblems by setting (i + -0.8846153845) equal to 2.256155289 and -2.256155289.Subproblem 1
i + -0.8846153845 = 2.256155289 Simplifying i + -0.8846153845 = 2.256155289 Reorder the terms: -0.8846153845 + i = 2.256155289 Solving -0.8846153845 + i = 2.256155289 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.8846153845' to each side of the equation. -0.8846153845 + 0.8846153845 + i = 2.256155289 + 0.8846153845 Combine like terms: -0.8846153845 + 0.8846153845 = 0.0000000000 0.0000000000 + i = 2.256155289 + 0.8846153845 i = 2.256155289 + 0.8846153845 Combine like terms: 2.256155289 + 0.8846153845 = 3.1407706735 i = 3.1407706735 Simplifying i = 3.1407706735Subproblem 2
i + -0.8846153845 = -2.256155289 Simplifying i + -0.8846153845 = -2.256155289 Reorder the terms: -0.8846153845 + i = -2.256155289 Solving -0.8846153845 + i = -2.256155289 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.8846153845' to each side of the equation. -0.8846153845 + 0.8846153845 + i = -2.256155289 + 0.8846153845 Combine like terms: -0.8846153845 + 0.8846153845 = 0.0000000000 0.0000000000 + i = -2.256155289 + 0.8846153845 i = -2.256155289 + 0.8846153845 Combine like terms: -2.256155289 + 0.8846153845 = -1.3715399045 i = -1.3715399045 Simplifying i = -1.3715399045Solution
The solution to the problem is based on the solutions from the subproblems. i = {3.1407706735, -1.3715399045}
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