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Simplifying 4(2 + -5i) + -1i(7 + 2i) = 0 (2 * 4 + -5i * 4) + -1i(7 + 2i) = 0 (8 + -20i) + -1i(7 + 2i) = 0 8 + -20i + (7 * -1i + 2i * -1i) = 0 8 + -20i + (-7i + -2i2) = 0 Combine like terms: -20i + -7i = -27i 8 + -27i + -2i2 = 0 Solving 8 + -27i + -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'. -4 + 13.5i + i2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 13.5i + 4 + i2 = 0 + 4 Reorder the terms: -4 + 4 + 13.5i + i2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 13.5i + i2 = 0 + 4 13.5i + i2 = 0 + 4 Combine like terms: 0 + 4 = 4 13.5i + i2 = 4 The i term is 13.5i. Take half its coefficient (6.75). Square it (45.5625) and add it to both sides. Add '45.5625' to each side of the equation. 13.5i + 45.5625 + i2 = 4 + 45.5625 Reorder the terms: 45.5625 + 13.5i + i2 = 4 + 45.5625 Combine like terms: 4 + 45.5625 = 49.5625 45.5625 + 13.5i + i2 = 49.5625 Factor a perfect square on the left side: (i + 6.75)(i + 6.75) = 49.5625 Calculate the square root of the right side: 7.04006392 Break this problem into two subproblems by setting (i + 6.75) equal to 7.04006392 and -7.04006392.Subproblem 1
i + 6.75 = 7.04006392 Simplifying i + 6.75 = 7.04006392 Reorder the terms: 6.75 + i = 7.04006392 Solving 6.75 + i = 7.04006392 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + i = 7.04006392 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + i = 7.04006392 + -6.75 i = 7.04006392 + -6.75 Combine like terms: 7.04006392 + -6.75 = 0.29006392 i = 0.29006392 Simplifying i = 0.29006392Subproblem 2
i + 6.75 = -7.04006392 Simplifying i + 6.75 = -7.04006392 Reorder the terms: 6.75 + i = -7.04006392 Solving 6.75 + i = -7.04006392 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + i = -7.04006392 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + i = -7.04006392 + -6.75 i = -7.04006392 + -6.75 Combine like terms: -7.04006392 + -6.75 = -13.79006392 i = -13.79006392 Simplifying i = -13.79006392Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.29006392, -13.79006392}
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