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Simplifying (11i2 + -7) + (2 + 4i) = 0 Reorder the terms: (-7 + 11i2) + (2 + 4i) = 0 Remove parenthesis around (-7 + 11i2) -7 + 11i2 + (2 + 4i) = 0 Remove parenthesis around (2 + 4i) -7 + 11i2 + 2 + 4i = 0 Reorder the terms: -7 + 2 + 4i + 11i2 = 0 Combine like terms: -7 + 2 = -5 -5 + 4i + 11i2 = 0 Solving -5 + 4i + 11i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.4545454545 + 0.3636363636i + i2 = 0 Move the constant term to the right: Add '0.4545454545' to each side of the equation. -0.4545454545 + 0.3636363636i + 0.4545454545 + i2 = 0 + 0.4545454545 Reorder the terms: -0.4545454545 + 0.4545454545 + 0.3636363636i + i2 = 0 + 0.4545454545 Combine like terms: -0.4545454545 + 0.4545454545 = 0.0000000000 0.0000000000 + 0.3636363636i + i2 = 0 + 0.4545454545 0.3636363636i + i2 = 0 + 0.4545454545 Combine like terms: 0 + 0.4545454545 = 0.4545454545 0.3636363636i + i2 = 0.4545454545 The i term is 0.3636363636i. Take half its coefficient (0.1818181818). Square it (0.03305785123) and add it to both sides. Add '0.03305785123' to each side of the equation. 0.3636363636i + 0.03305785123 + i2 = 0.4545454545 + 0.03305785123 Reorder the terms: 0.03305785123 + 0.3636363636i + i2 = 0.4545454545 + 0.03305785123 Combine like terms: 0.4545454545 + 0.03305785123 = 0.48760330573 0.03305785123 + 0.3636363636i + i2 = 0.48760330573 Factor a perfect square on the left side: (i + 0.1818181818)(i + 0.1818181818) = 0.48760330573 Calculate the square root of the right side: 0.698285977 Break this problem into two subproblems by setting (i + 0.1818181818) equal to 0.698285977 and -0.698285977.Subproblem 1
i + 0.1818181818 = 0.698285977 Simplifying i + 0.1818181818 = 0.698285977 Reorder the terms: 0.1818181818 + i = 0.698285977 Solving 0.1818181818 + i = 0.698285977 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.1818181818' to each side of the equation. 0.1818181818 + -0.1818181818 + i = 0.698285977 + -0.1818181818 Combine like terms: 0.1818181818 + -0.1818181818 = 0.0000000000 0.0000000000 + i = 0.698285977 + -0.1818181818 i = 0.698285977 + -0.1818181818 Combine like terms: 0.698285977 + -0.1818181818 = 0.5164677952 i = 0.5164677952 Simplifying i = 0.5164677952Subproblem 2
i + 0.1818181818 = -0.698285977 Simplifying i + 0.1818181818 = -0.698285977 Reorder the terms: 0.1818181818 + i = -0.698285977 Solving 0.1818181818 + i = -0.698285977 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.1818181818' to each side of the equation. 0.1818181818 + -0.1818181818 + i = -0.698285977 + -0.1818181818 Combine like terms: 0.1818181818 + -0.1818181818 = 0.0000000000 0.0000000000 + i = -0.698285977 + -0.1818181818 i = -0.698285977 + -0.1818181818 Combine like terms: -0.698285977 + -0.1818181818 = -0.8801041588 i = -0.8801041588 Simplifying i = -0.8801041588Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.5164677952, -0.8801041588}
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