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Simplifying 3[2 + -1(3j + -6)] * 4[6j + -1(1 + -2j)] = 4 + -5j Reorder the terms: 3[2 + -1(-6 + 3j)] * 4[6j + -1(1 + -2j)] = 4 + -5j 3[2 + (-6 * -1 + 3j * -1)] * 4[6j + -1(1 + -2j)] = 4 + -5j 3[2 + (6 + -3j)] * 4[6j + -1(1 + -2j)] = 4 + -5j Combine like terms: 2 + 6 = 8 3[8 + -3j] * 4[6j + -1(1 + -2j)] = 4 + -5j 3[8 + -3j] * 4[6j + (1 * -1 + -2j * -1)] = 4 + -5j 3[8 + -3j] * 4[6j + (-1 + 2j)] = 4 + -5j Reorder the terms: 3[8 + -3j] * 4[-1 + 6j + 2j] = 4 + -5j Combine like terms: 6j + 2j = 8j 3[8 + -3j] * 4[-1 + 8j] = 4 + -5j Reorder the terms for easier multiplication: 3 * 4[8 + -3j][-1 + 8j] = 4 + -5j Multiply 3 * 4 12[8 + -3j][-1 + 8j] = 4 + -5j Multiply [8 + -3j] * [-1 + 8j] 12[8[-1 + 8j] + -3j * [-1 + 8j]] = 4 + -5j 12[[-1 * 8 + 8j * 8] + -3j * [-1 + 8j]] = 4 + -5j 12[[-8 + 64j] + -3j * [-1 + 8j]] = 4 + -5j 12[-8 + 64j + [-1 * -3j + 8j * -3j]] = 4 + -5j 12[-8 + 64j + [3j + -24j2]] = 4 + -5j Combine like terms: 64j + 3j = 67j 12[-8 + 67j + -24j2] = 4 + -5j [-8 * 12 + 67j * 12 + -24j2 * 12] = 4 + -5j [-96 + 804j + -288j2] = 4 + -5j Solving -96 + 804j + -288j2 = 4 + -5j Solving for variable 'j'. Reorder the terms: -96 + -4 + 804j + 5j + -288j2 = 4 + -5j + -4 + 5j Combine like terms: -96 + -4 = -100 -100 + 804j + 5j + -288j2 = 4 + -5j + -4 + 5j Combine like terms: 804j + 5j = 809j -100 + 809j + -288j2 = 4 + -5j + -4 + 5j Reorder the terms: -100 + 809j + -288j2 = 4 + -4 + -5j + 5j Combine like terms: 4 + -4 = 0 -100 + 809j + -288j2 = 0 + -5j + 5j -100 + 809j + -288j2 = -5j + 5j Combine like terms: -5j + 5j = 0 -100 + 809j + -288j2 = 0 Begin completing the square. Divide all terms by -288 the coefficient of the squared term: Divide each side by '-288'. 0.3472222222 + -2.809027778j + j2 = 0 Move the constant term to the right: Add '-0.3472222222' to each side of the equation. 0.3472222222 + -2.809027778j + -0.3472222222 + j2 = 0 + -0.3472222222 Reorder the terms: 0.3472222222 + -0.3472222222 + -2.809027778j + j2 = 0 + -0.3472222222 Combine like terms: 0.3472222222 + -0.3472222222 = 0.0000000000 0.0000000000 + -2.809027778j + j2 = 0 + -0.3472222222 -2.809027778j + j2 = 0 + -0.3472222222 Combine like terms: 0 + -0.3472222222 = -0.3472222222 -2.809027778j + j2 = -0.3472222222 The j term is -2.809027778j. Take half its coefficient (-1.404513889). Square it (1.972659264) and add it to both sides. Add '1.972659264' to each side of the equation. -2.809027778j + 1.972659264 + j2 = -0.3472222222 + 1.972659264 Reorder the terms: 1.972659264 + -2.809027778j + j2 = -0.3472222222 + 1.972659264 Combine like terms: -0.3472222222 + 1.972659264 = 1.6254370418 1.972659264 + -2.809027778j + j2 = 1.6254370418 Factor a perfect square on the left side: (j + -1.404513889)(j + -1.404513889) = 1.6254370418 Calculate the square root of the right side: 1.274926289 Break this problem into two subproblems by setting (j + -1.404513889) equal to 1.274926289 and -1.274926289.Subproblem 1
j + -1.404513889 = 1.274926289 Simplifying j + -1.404513889 = 1.274926289 Reorder the terms: -1.404513889 + j = 1.274926289 Solving -1.404513889 + j = 1.274926289 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '1.404513889' to each side of the equation. -1.404513889 + 1.404513889 + j = 1.274926289 + 1.404513889 Combine like terms: -1.404513889 + 1.404513889 = 0.000000000 0.000000000 + j = 1.274926289 + 1.404513889 j = 1.274926289 + 1.404513889 Combine like terms: 1.274926289 + 1.404513889 = 2.679440178 j = 2.679440178 Simplifying j = 2.679440178Subproblem 2
j + -1.404513889 = -1.274926289 Simplifying j + -1.404513889 = -1.274926289 Reorder the terms: -1.404513889 + j = -1.274926289 Solving -1.404513889 + j = -1.274926289 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '1.404513889' to each side of the equation. -1.404513889 + 1.404513889 + j = -1.274926289 + 1.404513889 Combine like terms: -1.404513889 + 1.404513889 = 0.000000000 0.000000000 + j = -1.274926289 + 1.404513889 j = -1.274926289 + 1.404513889 Combine like terms: -1.274926289 + 1.404513889 = 0.1295876 j = 0.1295876 Simplifying j = 0.1295876Solution
The solution to the problem is based on the solutions from the subproblems. j = {2.679440178, 0.1295876}
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