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Simplifying 2j2 + -8 = -1j Reorder the terms: -8 + 2j2 = -1j Solving -8 + 2j2 = -1j Solving for variable 'j'. Reorder the terms: -8 + j + 2j2 = -1j + j Combine like terms: -1j + j = 0 -8 + j + 2j2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4 + 0.5j + j2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.5j + 4 + j2 = 0 + 4 Reorder the terms: -4 + 4 + 0.5j + j2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.5j + j2 = 0 + 4 0.5j + j2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.5j + j2 = 4 The j term is j. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5j + 0.25 + j2 = 4 + 0.25 Reorder the terms: 0.25 + 0.5j + j2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + 0.5j + j2 = 4.25 Factor a perfect square on the left side: (j + 0.5)(j + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (j + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
j + 0.5 = 2.061552813 Simplifying j + 0.5 = 2.061552813 Reorder the terms: 0.5 + j = 2.061552813 Solving 0.5 + j = 2.061552813 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + j = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + j = 2.061552813 + -0.5 j = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 j = 1.561552813 Simplifying j = 1.561552813Subproblem 2
j + 0.5 = -2.061552813 Simplifying j + 0.5 = -2.061552813 Reorder the terms: 0.5 + j = -2.061552813 Solving 0.5 + j = -2.061552813 Solving for variable 'j'. Move all terms containing j to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + j = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + j = -2.061552813 + -0.5 j = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 j = -2.561552813 Simplifying j = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. j = {1.561552813, -2.561552813}
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