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Simplifying (5 + 8j)(5 + j) = 1 Multiply (5 + 8j) * (5 + j) (5(5 + j) + 8j * (5 + j)) = 1 ((5 * 5 + j * 5) + 8j * (5 + j)) = 1 ((25 + 5j) + 8j * (5 + j)) = 1 (25 + 5j + (5 * 8j + j * 8j)) = 1 (25 + 5j + (40j + 8j2)) = 1 Combine like terms: 5j + 40j = 45j (25 + 45j + 8j2) = 1 Solving 25 + 45j + 8j2 = 1 Solving for variable 'j'. Reorder the terms: 25 + -1 + 45j + 8j2 = 1 + -1 Combine like terms: 25 + -1 = 24 24 + 45j + 8j2 = 1 + -1 Combine like terms: 1 + -1 = 0 24 + 45j + 8j2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3 + 5.625j + j2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + 5.625j + -3 + j2 = 0 + -3 Reorder the terms: 3 + -3 + 5.625j + j2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 5.625j + j2 = 0 + -3 5.625j + j2 = 0 + -3 Combine like terms: 0 + -3 = -3 5.625j + j2 = -3 The j term is 5.625j. Take half its coefficient (2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. 5.625j + 7.91015625 + j2 = -3 + 7.91015625 Reorder the terms: 7.91015625 + 5.625j + j2 = -3 + 7.91015625 Combine like terms: -3 + 7.91015625 = 4.91015625 7.91015625 + 5.625j + j2 = 4.91015625 Factor a perfect square on the left side: (j + 2.8125)(j + 2.8125) = 4.91015625 Calculate the square root of the right side: 2.215887238 Break this problem into two subproblems by setting (j + 2.8125) equal to 2.215887238 and -2.215887238.Subproblem 1
j + 2.8125 = 2.215887238 Simplifying j + 2.8125 = 2.215887238 Reorder the terms: 2.8125 + j = 2.215887238 Solving 2.8125 + j = 2.215887238 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-2.8125' to each side of the equation. 2.8125 + -2.8125 + j = 2.215887238 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + j = 2.215887238 + -2.8125 j = 2.215887238 + -2.8125 Combine like terms: 2.215887238 + -2.8125 = -0.596612762 j = -0.596612762 Simplifying j = -0.596612762Subproblem 2
j + 2.8125 = -2.215887238 Simplifying j + 2.8125 = -2.215887238 Reorder the terms: 2.8125 + j = -2.215887238 Solving 2.8125 + j = -2.215887238 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-2.8125' to each side of the equation. 2.8125 + -2.8125 + j = -2.215887238 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + j = -2.215887238 + -2.8125 j = -2.215887238 + -2.8125 Combine like terms: -2.215887238 + -2.8125 = -5.028387238 j = -5.028387238 Simplifying j = -5.028387238Solution
The solution to the problem is based on the solutions from the subproblems. j = {-0.596612762, -5.028387238}
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