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Simplifying (1 + 5i)(-1i) + -1(4 + 3i)(4 + -3i) = 0 Remove parenthesis around (-1i) (1 + 5i) * -1i + -1(4 + 3i)(4 + -3i) = 0 Reorder the terms for easier multiplication: -1i(1 + 5i) + -1(4 + 3i)(4 + -3i) = 0 (1 * -1i + 5i * -1i) + -1(4 + 3i)(4 + -3i) = 0 (-1i + -5i2) + -1(4 + 3i)(4 + -3i) = 0 Multiply (4 + 3i) * (4 + -3i) -1i + -5i2 + -1(4(4 + -3i) + 3i * (4 + -3i)) = 0 -1i + -5i2 + -1((4 * 4 + -3i * 4) + 3i * (4 + -3i)) = 0 -1i + -5i2 + -1((16 + -12i) + 3i * (4 + -3i)) = 0 -1i + -5i2 + -1(16 + -12i + (4 * 3i + -3i * 3i)) = 0 -1i + -5i2 + -1(16 + -12i + (12i + -9i2)) = 0 Combine like terms: -12i + 12i = 0 -1i + -5i2 + -1(16 + 0 + -9i2) = 0 -1i + -5i2 + -1(16 + -9i2) = 0 -1i + -5i2 + (16 * -1 + -9i2 * -1) = 0 -1i + -5i2 + (-16 + 9i2) = 0 Reorder the terms: -16 + -1i + -5i2 + 9i2 = 0 Combine like terms: -5i2 + 9i2 = 4i2 -16 + -1i + 4i2 = 0 Solving -16 + -1i + 4i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4 + -0.25i + i2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -0.25i + 4 + i2 = 0 + 4 Reorder the terms: -4 + 4 + -0.25i + i2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -0.25i + i2 = 0 + 4 -0.25i + i2 = 0 + 4 Combine like terms: 0 + 4 = 4 -0.25i + i2 = 4 The i term is -0.25i. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25i + 0.015625 + i2 = 4 + 0.015625 Reorder the terms: 0.015625 + -0.25i + i2 = 4 + 0.015625 Combine like terms: 4 + 0.015625 = 4.015625 0.015625 + -0.25i + i2 = 4.015625 Factor a perfect square on the left side: (i + -0.125)(i + -0.125) = 4.015625 Calculate the square root of the right side: 2.003902443 Break this problem into two subproblems by setting (i + -0.125) equal to 2.003902443 and -2.003902443.Subproblem 1
i + -0.125 = 2.003902443 Simplifying i + -0.125 = 2.003902443 Reorder the terms: -0.125 + i = 2.003902443 Solving -0.125 + i = 2.003902443 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + i = 2.003902443 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + i = 2.003902443 + 0.125 i = 2.003902443 + 0.125 Combine like terms: 2.003902443 + 0.125 = 2.128902443 i = 2.128902443 Simplifying i = 2.128902443Subproblem 2
i + -0.125 = -2.003902443 Simplifying i + -0.125 = -2.003902443 Reorder the terms: -0.125 + i = -2.003902443 Solving -0.125 + i = -2.003902443 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + i = -2.003902443 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + i = -2.003902443 + 0.125 i = -2.003902443 + 0.125 Combine like terms: -2.003902443 + 0.125 = -1.878902443 i = -1.878902443 Simplifying i = -1.878902443Solution
The solution to the problem is based on the solutions from the subproblems. i = {2.128902443, -1.878902443}
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