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Simplifying -1i(8 + -2i) + -1(5 + -9i) = 0 (8 * -1i + -2i * -1i) + -1(5 + -9i) = 0 (-8i + 2i2) + -1(5 + -9i) = 0 -8i + 2i2 + (5 * -1 + -9i * -1) = 0 -8i + 2i2 + (-5 + 9i) = 0 Reorder the terms: -5 + -8i + 9i + 2i2 = 0 Combine like terms: -8i + 9i = 1i -5 + 1i + 2i2 = 0 Solving -5 + 1i + 2i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 0.5i + i2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.5i + 2.5 + i2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.5i + i2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.5i + i2 = 0 + 2.5 0.5i + i2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 0.5i + i2 = 2.5 The i term is 0.5i. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5i + 0.0625 + i2 = 2.5 + 0.0625 Reorder the terms: 0.0625 + 0.5i + i2 = 2.5 + 0.0625 Combine like terms: 2.5 + 0.0625 = 2.5625 0.0625 + 0.5i + i2 = 2.5625 Factor a perfect square on the left side: (i + 0.25)(i + 0.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (i + 0.25) equal to 1.600781059 and -1.600781059.Subproblem 1
i + 0.25 = 1.600781059 Simplifying i + 0.25 = 1.600781059 Reorder the terms: 0.25 + i = 1.600781059 Solving 0.25 + i = 1.600781059 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + i = 1.600781059 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + i = 1.600781059 + -0.25 i = 1.600781059 + -0.25 Combine like terms: 1.600781059 + -0.25 = 1.350781059 i = 1.350781059 Simplifying i = 1.350781059Subproblem 2
i + 0.25 = -1.600781059 Simplifying i + 0.25 = -1.600781059 Reorder the terms: 0.25 + i = -1.600781059 Solving 0.25 + i = -1.600781059 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + i = -1.600781059 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + i = -1.600781059 + -0.25 i = -1.600781059 + -0.25 Combine like terms: -1.600781059 + -0.25 = -1.850781059 i = -1.850781059 Simplifying i = -1.850781059Solution
The solution to the problem is based on the solutions from the subproblems. i = {1.350781059, -1.850781059}
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