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Simplifying (16i + -9i)(-10 + -5i) + (4 + -2i) = 0 Combine like terms: 16i + -9i = 7i (7i)(-10 + -5i) + (4 + -2i) = 0 Remove parenthesis around (7i) 7i(-10 + -5i) + (4 + -2i) = 0 (-10 * 7i + -5i * 7i) + (4 + -2i) = 0 (-70i + -35i2) + (4 + -2i) = 0 Remove parenthesis around (4 + -2i) -70i + -35i2 + 4 + -2i = 0 Reorder the terms: 4 + -70i + -2i + -35i2 = 0 Combine like terms: -70i + -2i = -72i 4 + -72i + -35i2 = 0 Solving 4 + -72i + -35i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -35 the coefficient of the squared term: Divide each side by '-35'. -0.1142857143 + 2.057142857i + i2 = 0 Move the constant term to the right: Add '0.1142857143' to each side of the equation. -0.1142857143 + 2.057142857i + 0.1142857143 + i2 = 0 + 0.1142857143 Reorder the terms: -0.1142857143 + 0.1142857143 + 2.057142857i + i2 = 0 + 0.1142857143 Combine like terms: -0.1142857143 + 0.1142857143 = 0.0000000000 0.0000000000 + 2.057142857i + i2 = 0 + 0.1142857143 2.057142857i + i2 = 0 + 0.1142857143 Combine like terms: 0 + 0.1142857143 = 0.1142857143 2.057142857i + i2 = 0.1142857143 The i term is 2.057142857i. Take half its coefficient (1.028571429). Square it (1.057959185) and add it to both sides. Add '1.057959185' to each side of the equation. 2.057142857i + 1.057959185 + i2 = 0.1142857143 + 1.057959185 Reorder the terms: 1.057959185 + 2.057142857i + i2 = 0.1142857143 + 1.057959185 Combine like terms: 0.1142857143 + 1.057959185 = 1.1722448993 1.057959185 + 2.057142857i + i2 = 1.1722448993 Factor a perfect square on the left side: (i + 1.028571429)(i + 1.028571429) = 1.1722448993 Calculate the square root of the right side: 1.08270259 Break this problem into two subproblems by setting (i + 1.028571429) equal to 1.08270259 and -1.08270259.Subproblem 1
i + 1.028571429 = 1.08270259 Simplifying i + 1.028571429 = 1.08270259 Reorder the terms: 1.028571429 + i = 1.08270259 Solving 1.028571429 + i = 1.08270259 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.028571429' to each side of the equation. 1.028571429 + -1.028571429 + i = 1.08270259 + -1.028571429 Combine like terms: 1.028571429 + -1.028571429 = 0.000000000 0.000000000 + i = 1.08270259 + -1.028571429 i = 1.08270259 + -1.028571429 Combine like terms: 1.08270259 + -1.028571429 = 0.054131161 i = 0.054131161 Simplifying i = 0.054131161Subproblem 2
i + 1.028571429 = -1.08270259 Simplifying i + 1.028571429 = -1.08270259 Reorder the terms: 1.028571429 + i = -1.08270259 Solving 1.028571429 + i = -1.08270259 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.028571429' to each side of the equation. 1.028571429 + -1.028571429 + i = -1.08270259 + -1.028571429 Combine like terms: 1.028571429 + -1.028571429 = 0.000000000 0.000000000 + i = -1.08270259 + -1.028571429 i = -1.08270259 + -1.028571429 Combine like terms: -1.08270259 + -1.028571429 = -2.111274019 i = -2.111274019 Simplifying i = -2.111274019Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.054131161, -2.111274019}
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