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Simplifying (-2 + -1j) + (-3 + -1j)(0.707 + -0.707j) = 0 Remove parenthesis around (-2 + -1j) -2 + -1j + (-3 + -1j)(0.707 + -0.707j) = 0 Multiply (-3 + -1j) * (0.707 + -0.707j) -2 + -1j + (-3(0.707 + -0.707j) + -1j * (0.707 + -0.707j)) = 0 -2 + -1j + ((0.707 * -3 + -0.707j * -3) + -1j * (0.707 + -0.707j)) = 0 -2 + -1j + ((-2.121 + 2.121j) + -1j * (0.707 + -0.707j)) = 0 -2 + -1j + (-2.121 + 2.121j + (0.707 * -1j + -0.707j * -1j)) = 0 -2 + -1j + (-2.121 + 2.121j + (-0.707j + 0.707j2)) = 0 Combine like terms: 2.121j + -0.707j = 1.414j -2 + -1j + (-2.121 + 1.414j + 0.707j2) = 0 Reorder the terms: -2 + -2.121 + -1j + 1.414j + 0.707j2 = 0 Combine like terms: -2 + -2.121 = -4.121 -4.121 + -1j + 1.414j + 0.707j2 = 0 Combine like terms: -1j + 1.414j = 0.414j -4.121 + 0.414j + 0.707j2 = 0 Solving -4.121 + 0.414j + 0.707j2 = 0 Solving for variable 'j'. Begin completing the square. Divide all terms by 0.707 the coefficient of the squared term: Divide each side by '0.707'. -5.828854314 + 0.585572843j + j2 = 0 Move the constant term to the right: Add '5.828854314' to each side of the equation. -5.828854314 + 0.585572843j + 5.828854314 + j2 = 0 + 5.828854314 Reorder the terms: -5.828854314 + 5.828854314 + 0.585572843j + j2 = 0 + 5.828854314 Combine like terms: -5.828854314 + 5.828854314 = 0.000000000 0.000000000 + 0.585572843j + j2 = 0 + 5.828854314 0.585572843j + j2 = 0 + 5.828854314 Combine like terms: 0 + 5.828854314 = 5.828854314 0.585572843j + j2 = 5.828854314 The j term is 0.585572843j. Take half its coefficient (0.2927864215). Square it (0.08572388861) and add it to both sides. Add '0.08572388861' to each side of the equation. 0.585572843j + 0.08572388861 + j2 = 5.828854314 + 0.08572388861 Reorder the terms: 0.08572388861 + 0.585572843j + j2 = 5.828854314 + 0.08572388861 Combine like terms: 5.828854314 + 0.08572388861 = 5.91457820261 0.08572388861 + 0.585572843j + j2 = 5.91457820261 Factor a perfect square on the left side: (j + 0.2927864215)(j + 0.2927864215) = 5.91457820261 Calculate the square root of the right side: 2.431990584 Break this problem into two subproblems by setting (j + 0.2927864215) equal to 2.431990584 and -2.431990584.Subproblem 1
j + 0.2927864215 = 2.431990584 Simplifying j + 0.2927864215 = 2.431990584 Reorder the terms: 0.2927864215 + j = 2.431990584 Solving 0.2927864215 + j = 2.431990584 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.2927864215' to each side of the equation. 0.2927864215 + -0.2927864215 + j = 2.431990584 + -0.2927864215 Combine like terms: 0.2927864215 + -0.2927864215 = 0.0000000000 0.0000000000 + j = 2.431990584 + -0.2927864215 j = 2.431990584 + -0.2927864215 Combine like terms: 2.431990584 + -0.2927864215 = 2.1392041625 j = 2.1392041625 Simplifying j = 2.1392041625Subproblem 2
j + 0.2927864215 = -2.431990584 Simplifying j + 0.2927864215 = -2.431990584 Reorder the terms: 0.2927864215 + j = -2.431990584 Solving 0.2927864215 + j = -2.431990584 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.2927864215' to each side of the equation. 0.2927864215 + -0.2927864215 + j = -2.431990584 + -0.2927864215 Combine like terms: 0.2927864215 + -0.2927864215 = 0.0000000000 0.0000000000 + j = -2.431990584 + -0.2927864215 j = -2.431990584 + -0.2927864215 Combine like terms: -2.431990584 + -0.2927864215 = -2.7247770055 j = -2.7247770055 Simplifying j = -2.7247770055Solution
The solution to the problem is based on the solutions from the subproblems. j = {2.1392041625, -2.7247770055}
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