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Simplifying (2(-1j)) + (-3 + j)(-0.707 + -0.707j) = 0 Remove parenthesis around (-1j) (2 * -1j) + (-3 + j)(-0.707 + -0.707j) = 0 Multiply 2 * -1 (-2j) + (-3 + j)(-0.707 + -0.707j) = 0 Multiply (-3 + j) * (-0.707 + -0.707j) (-2j) + (-3(-0.707 + -0.707j) + j(-0.707 + -0.707j)) = 0 (-2j) + ((-0.707 * -3 + -0.707j * -3) + j(-0.707 + -0.707j)) = 0 (-2j) + ((2.121 + 2.121j) + j(-0.707 + -0.707j)) = 0 (-2j) + (2.121 + 2.121j + (-0.707 * j + -0.707j * j)) = 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 = -0.586j 2.121 + -0.586j + -0.707j2 = 0 Solving 2.121 + -0.586j + -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 + 0.828854314j + j2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.828854314j + 3 + j2 = 0 + 3 Reorder the terms: -3 + 3 + 0.828854314j + j2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.828854314j + j2 = 0 + 3 0.828854314j + j2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.828854314j + j2 = 3 The j term is 0.828854314j. Take half its coefficient (0.414427157). Square it (0.1717498685) and add it to both sides. Add '0.1717498685' to each side of the equation. 0.828854314j + 0.1717498685 + j2 = 3 + 0.1717498685 Reorder the terms: 0.1717498685 + 0.828854314j + j2 = 3 + 0.1717498685 Combine like terms: 3 + 0.1717498685 = 3.1717498685 0.1717498685 + 0.828854314j + j2 = 3.1717498685 Factor a perfect square on the left side: (j + 0.414427157)(j + 0.414427157) = 3.1717498685 Calculate the square root of the right side: 1.780940726 Break this problem into two subproblems by setting (j + 0.414427157) equal to 1.780940726 and -1.780940726.Subproblem 1
j + 0.414427157 = 1.780940726 Simplifying j + 0.414427157 = 1.780940726 Reorder the terms: 0.414427157 + j = 1.780940726 Solving 0.414427157 + j = 1.780940726 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.414427157' to each side of the equation. 0.414427157 + -0.414427157 + j = 1.780940726 + -0.414427157 Combine like terms: 0.414427157 + -0.414427157 = 0.000000000 0.000000000 + j = 1.780940726 + -0.414427157 j = 1.780940726 + -0.414427157 Combine like terms: 1.780940726 + -0.414427157 = 1.366513569 j = 1.366513569 Simplifying j = 1.366513569Subproblem 2
j + 0.414427157 = -1.780940726 Simplifying j + 0.414427157 = -1.780940726 Reorder the terms: 0.414427157 + j = -1.780940726 Solving 0.414427157 + j = -1.780940726 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.414427157' to each side of the equation. 0.414427157 + -0.414427157 + j = -1.780940726 + -0.414427157 Combine like terms: 0.414427157 + -0.414427157 = 0.000000000 0.000000000 + j = -1.780940726 + -0.414427157 j = -1.780940726 + -0.414427157 Combine like terms: -1.780940726 + -0.414427157 = -2.195367883 j = -2.195367883 Simplifying j = -2.195367883Solution
The solution to the problem is based on the solutions from the subproblems. j = {1.366513569, -2.195367883}
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