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Simplifying (4 + 2j) + -1(-4 + -2j) + -1(7 + -1j) + j(5 + -4j) = 0 Remove parenthesis around (4 + 2j) 4 + 2j + -1(-4 + -2j) + -1(7 + -1j) + j(5 + -4j) = 0 4 + 2j + (-4 * -1 + -2j * -1) + -1(7 + -1j) + j(5 + -4j) = 0 4 + 2j + (4 + 2j) + -1(7 + -1j) + j(5 + -4j) = 0 4 + 2j + 4 + 2j + (7 * -1 + -1j * -1) + j(5 + -4j) = 0 4 + 2j + 4 + 2j + (-7 + 1j) + j(5 + -4j) = 0 4 + 2j + 4 + 2j + -7 + 1j + (5 * j + -4j * j) = 0 4 + 2j + 4 + 2j + -7 + 1j + (5j + -4j2) = 0 Reorder the terms: 4 + 4 + -7 + 2j + 2j + 1j + 5j + -4j2 = 0 Combine like terms: 4 + 4 = 8 8 + -7 + 2j + 2j + 1j + 5j + -4j2 = 0 Combine like terms: 8 + -7 = 1 1 + 2j + 2j + 1j + 5j + -4j2 = 0 Combine like terms: 2j + 2j = 4j 1 + 4j + 1j + 5j + -4j2 = 0 Combine like terms: 4j + 1j = 5j 1 + 5j + 5j + -4j2 = 0 Combine like terms: 5j + 5j = 10j 1 + 10j + -4j2 = 0 Solving 1 + 10j + -4j2 = 0 Solving for variable 'j'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -2.5j + j2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -2.5j + 0.25 + j2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -2.5j + j2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -2.5j + j2 = 0 + 0.25 -2.5j + j2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -2.5j + j2 = 0.25 The j term is -2.5j. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5j + 1.5625 + j2 = 0.25 + 1.5625 Reorder the terms: 1.5625 + -2.5j + j2 = 0.25 + 1.5625 Combine like terms: 0.25 + 1.5625 = 1.8125 1.5625 + -2.5j + j2 = 1.8125 Factor a perfect square on the left side: (j + -1.25)(j + -1.25) = 1.8125 Calculate the square root of the right side: 1.346291202 Break this problem into two subproblems by setting (j + -1.25) equal to 1.346291202 and -1.346291202.Subproblem 1
j + -1.25 = 1.346291202 Simplifying j + -1.25 = 1.346291202 Reorder the terms: -1.25 + j = 1.346291202 Solving -1.25 + j = 1.346291202 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + j = 1.346291202 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + j = 1.346291202 + 1.25 j = 1.346291202 + 1.25 Combine like terms: 1.346291202 + 1.25 = 2.596291202 j = 2.596291202 Simplifying j = 2.596291202Subproblem 2
j + -1.25 = -1.346291202 Simplifying j + -1.25 = -1.346291202 Reorder the terms: -1.25 + j = -1.346291202 Solving -1.25 + j = -1.346291202 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + j = -1.346291202 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + j = -1.346291202 + 1.25 j = -1.346291202 + 1.25 Combine like terms: -1.346291202 + 1.25 = -0.096291202 j = -0.096291202 Simplifying j = -0.096291202Solution
The solution to the problem is based on the solutions from the subproblems. j = {2.596291202, -0.096291202}
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