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Simplifying 4(2j + 1) + -1j(2 + -1j) = 0 Reorder the terms: 4(1 + 2j) + -1j(2 + -1j) = 0 (1 * 4 + 2j * 4) + -1j(2 + -1j) = 0 (4 + 8j) + -1j(2 + -1j) = 0 4 + 8j + (2 * -1j + -1j * -1j) = 0 4 + 8j + (-2j + 1j2) = 0 Combine like terms: 8j + -2j = 6j 4 + 6j + 1j2 = 0 Solving 4 + 6j + 1j2 = 0 Solving for variable 'j'. Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + 6j + -4 + j2 = 0 + -4 Reorder the terms: 4 + -4 + 6j + j2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + 6j + j2 = 0 + -4 6j + j2 = 0 + -4 Combine like terms: 0 + -4 = -4 6j + j2 = -4 The j term is 6j. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6j + 9 + j2 = -4 + 9 Reorder the terms: 9 + 6j + j2 = -4 + 9 Combine like terms: -4 + 9 = 5 9 + 6j + j2 = 5 Factor a perfect square on the left side: (j + 3)(j + 3) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (j + 3) equal to 2.236067978 and -2.236067978.Subproblem 1
j + 3 = 2.236067978 Simplifying j + 3 = 2.236067978 Reorder the terms: 3 + j = 2.236067978 Solving 3 + j = 2.236067978 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + j = 2.236067978 + -3 Combine like terms: 3 + -3 = 0 0 + j = 2.236067978 + -3 j = 2.236067978 + -3 Combine like terms: 2.236067978 + -3 = -0.763932022 j = -0.763932022 Simplifying j = -0.763932022Subproblem 2
j + 3 = -2.236067978 Simplifying j + 3 = -2.236067978 Reorder the terms: 3 + j = -2.236067978 Solving 3 + j = -2.236067978 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + j = -2.236067978 + -3 Combine like terms: 3 + -3 = 0 0 + j = -2.236067978 + -3 j = -2.236067978 + -3 Combine like terms: -2.236067978 + -3 = -5.236067978 j = -5.236067978 Simplifying j = -5.236067978Solution
The solution to the problem is based on the solutions from the subproblems. j = {-0.763932022, -5.236067978}
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