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Simplifying j + -15j + 8 = -3(4j + -3) * 2j + -1 Reorder the terms: 8 + j + -15j = -3(4j + -3) * 2j + -1 Combine like terms: j + -15j = -14j 8 + -14j = -3(4j + -3) * 2j + -1 Reorder the terms: 8 + -14j = -3(-3 + 4j) * 2j + -1 Reorder the terms for easier multiplication: 8 + -14j = -3 * 2j(-3 + 4j) + -1 Multiply -3 * 2 8 + -14j = -6j(-3 + 4j) + -1 8 + -14j = (-3 * -6j + 4j * -6j) + -1 8 + -14j = (18j + -24j2) + -1 Reorder the terms: 8 + -14j = -1 + 18j + -24j2 Solving 8 + -14j = -1 + 18j + -24j2 Solving for variable 'j'. Reorder the terms: 8 + 1 + -14j + -18j + 24j2 = -1 + 18j + -24j2 + 1 + -18j + 24j2 Combine like terms: 8 + 1 = 9 9 + -14j + -18j + 24j2 = -1 + 18j + -24j2 + 1 + -18j + 24j2 Combine like terms: -14j + -18j = -32j 9 + -32j + 24j2 = -1 + 18j + -24j2 + 1 + -18j + 24j2 Reorder the terms: 9 + -32j + 24j2 = -1 + 1 + 18j + -18j + -24j2 + 24j2 Combine like terms: -1 + 1 = 0 9 + -32j + 24j2 = 0 + 18j + -18j + -24j2 + 24j2 9 + -32j + 24j2 = 18j + -18j + -24j2 + 24j2 Combine like terms: 18j + -18j = 0 9 + -32j + 24j2 = 0 + -24j2 + 24j2 9 + -32j + 24j2 = -24j2 + 24j2 Combine like terms: -24j2 + 24j2 = 0 9 + -32j + 24j2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 0.375 + -1.333333333j + j2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + -1.333333333j + -0.375 + j2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + -1.333333333j + j2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + -1.333333333j + j2 = 0 + -0.375 -1.333333333j + j2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 -1.333333333j + j2 = -0.375 The j term is -1.333333333j. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333j + 0.4444444442 + j2 = -0.375 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333j + j2 = -0.375 + 0.4444444442 Combine like terms: -0.375 + 0.4444444442 = 0.0694444442 0.4444444442 + -1.333333333j + j2 = 0.0694444442 Factor a perfect square on the left side: (j + -0.6666666665)(j + -0.6666666665) = 0.0694444442 Calculate the square root of the right side: 0.263523138 Break this problem into two subproblems by setting (j + -0.6666666665) equal to 0.263523138 and -0.263523138.Subproblem 1
j + -0.6666666665 = 0.263523138 Simplifying j + -0.6666666665 = 0.263523138 Reorder the terms: -0.6666666665 + j = 0.263523138 Solving -0.6666666665 + j = 0.263523138 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + j = 0.263523138 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + j = 0.263523138 + 0.6666666665 j = 0.263523138 + 0.6666666665 Combine like terms: 0.263523138 + 0.6666666665 = 0.9301898045 j = 0.9301898045 Simplifying j = 0.9301898045Subproblem 2
j + -0.6666666665 = -0.263523138 Simplifying j + -0.6666666665 = -0.263523138 Reorder the terms: -0.6666666665 + j = -0.263523138 Solving -0.6666666665 + j = -0.263523138 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + j = -0.263523138 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + j = -0.263523138 + 0.6666666665 j = -0.263523138 + 0.6666666665 Combine like terms: -0.263523138 + 0.6666666665 = 0.4031435285 j = 0.4031435285 Simplifying j = 0.4031435285Solution
The solution to the problem is based on the solutions from the subproblems. j = {0.9301898045, 0.4031435285}
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