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Simplifying (6 + 3j)(7 + j) = 1 Multiply (6 + 3j) * (7 + j) (6(7 + j) + 3j * (7 + j)) = 1 ((7 * 6 + j * 6) + 3j * (7 + j)) = 1 ((42 + 6j) + 3j * (7 + j)) = 1 (42 + 6j + (7 * 3j + j * 3j)) = 1 (42 + 6j + (21j + 3j2)) = 1 Combine like terms: 6j + 21j = 27j (42 + 27j + 3j2) = 1 Solving 42 + 27j + 3j2 = 1 Solving for variable 'j'. Reorder the terms: 42 + -1 + 27j + 3j2 = 1 + -1 Combine like terms: 42 + -1 = 41 41 + 27j + 3j2 = 1 + -1 Combine like terms: 1 + -1 = 0 41 + 27j + 3j2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 13.66666667 + 9j + j2 = 0 Move the constant term to the right: Add '-13.66666667' to each side of the equation. 13.66666667 + 9j + -13.66666667 + j2 = 0 + -13.66666667 Reorder the terms: 13.66666667 + -13.66666667 + 9j + j2 = 0 + -13.66666667 Combine like terms: 13.66666667 + -13.66666667 = 0.00000000 0.00000000 + 9j + j2 = 0 + -13.66666667 9j + j2 = 0 + -13.66666667 Combine like terms: 0 + -13.66666667 = -13.66666667 9j + j2 = -13.66666667 The j term is 9j. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9j + 20.25 + j2 = -13.66666667 + 20.25 Reorder the terms: 20.25 + 9j + j2 = -13.66666667 + 20.25 Combine like terms: -13.66666667 + 20.25 = 6.58333333 20.25 + 9j + j2 = 6.58333333 Factor a perfect square on the left side: (j + 4.5)(j + 4.5) = 6.58333333 Calculate the square root of the right side: 2.565800719 Break this problem into two subproblems by setting (j + 4.5) equal to 2.565800719 and -2.565800719.Subproblem 1
j + 4.5 = 2.565800719 Simplifying j + 4.5 = 2.565800719 Reorder the terms: 4.5 + j = 2.565800719 Solving 4.5 + j = 2.565800719 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + j = 2.565800719 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + j = 2.565800719 + -4.5 j = 2.565800719 + -4.5 Combine like terms: 2.565800719 + -4.5 = -1.934199281 j = -1.934199281 Simplifying j = -1.934199281Subproblem 2
j + 4.5 = -2.565800719 Simplifying j + 4.5 = -2.565800719 Reorder the terms: 4.5 + j = -2.565800719 Solving 4.5 + j = -2.565800719 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + j = -2.565800719 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + j = -2.565800719 + -4.5 j = -2.565800719 + -4.5 Combine like terms: -2.565800719 + -4.5 = -7.065800719 j = -7.065800719 Simplifying j = -7.065800719Solution
The solution to the problem is based on the solutions from the subproblems. j = {-1.934199281, -7.065800719}
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