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Simplifying (-2(-1j)) + (-3 + -1j)(0.707 + -0.707j) = 0 Remove parenthesis around (-1j) (-2 * -1j) + (-3 + -1j)(0.707 + -0.707j) = 0 Multiply -2 * -1 (2j) + (-3 + -1j)(0.707 + -0.707j) = 0 Multiply (-3 + -1j) * (0.707 + -0.707j) (2j) + (-3(0.707 + -0.707j) + -1j * (0.707 + -0.707j)) = 0 (2j) + ((0.707 * -3 + -0.707j * -3) + -1j * (0.707 + -0.707j)) = 0 (2j) + ((-2.121 + 2.121j) + -1j * (0.707 + -0.707j)) = 0 (2j) + (-2.121 + 2.121j + (0.707 * -1j + -0.707j * -1j)) = 0 (2j) + (-2.121 + 2.121j + (-0.707j + 0.707j2)) = 0 Combine like terms: 2.121j + -0.707j = 1.414j (2j) + (-2.121 + 1.414j + 0.707j2) = 0 Reorder the terms: -2.121 + (2j) + 1.414j + 0.707j2 = 0 Combine like terms: (2j) + 1.414j = 3.414j -2.121 + 3.414j + 0.707j2 = 0 Solving -2.121 + 3.414j + 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'. -3 + 4.828854314j + j2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 4.828854314j + 3 + j2 = 0 + 3 Reorder the terms: -3 + 3 + 4.828854314j + j2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 4.828854314j + j2 = 0 + 3 4.828854314j + j2 = 0 + 3 Combine like terms: 0 + 3 = 3 4.828854314j + j2 = 3 The j term is 4.828854314j. Take half its coefficient (2.414427157). Square it (5.829458496) and add it to both sides. Add '5.829458496' to each side of the equation. 4.828854314j + 5.829458496 + j2 = 3 + 5.829458496 Reorder the terms: 5.829458496 + 4.828854314j + j2 = 3 + 5.829458496 Combine like terms: 3 + 5.829458496 = 8.829458496 5.829458496 + 4.828854314j + j2 = 8.829458496 Factor a perfect square on the left side: (j + 2.414427157)(j + 2.414427157) = 8.829458496 Calculate the square root of the right side: 2.971440475 Break this problem into two subproblems by setting (j + 2.414427157) equal to 2.971440475 and -2.971440475.Subproblem 1
j + 2.414427157 = 2.971440475 Simplifying j + 2.414427157 = 2.971440475 Reorder the terms: 2.414427157 + j = 2.971440475 Solving 2.414427157 + j = 2.971440475 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-2.414427157' to each side of the equation. 2.414427157 + -2.414427157 + j = 2.971440475 + -2.414427157 Combine like terms: 2.414427157 + -2.414427157 = 0.000000000 0.000000000 + j = 2.971440475 + -2.414427157 j = 2.971440475 + -2.414427157 Combine like terms: 2.971440475 + -2.414427157 = 0.557013318 j = 0.557013318 Simplifying j = 0.557013318Subproblem 2
j + 2.414427157 = -2.971440475 Simplifying j + 2.414427157 = -2.971440475 Reorder the terms: 2.414427157 + j = -2.971440475 Solving 2.414427157 + j = -2.971440475 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-2.414427157' to each side of the equation. 2.414427157 + -2.414427157 + j = -2.971440475 + -2.414427157 Combine like terms: 2.414427157 + -2.414427157 = 0.000000000 0.000000000 + j = -2.971440475 + -2.414427157 j = -2.971440475 + -2.414427157 Combine like terms: -2.971440475 + -2.414427157 = -5.385867632 j = -5.385867632 Simplifying j = -5.385867632Solution
The solution to the problem is based on the solutions from the subproblems. j = {0.557013318, -5.385867632}
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