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Simplifying (i + 1)(3 + -1i) + (2i + -1) = 0 Reorder the terms: (1 + i)(3 + -1i) + (2i + -1) = 0 Multiply (1 + i) * (3 + -1i) (1(3 + -1i) + i(3 + -1i)) + (2i + -1) = 0 ((3 * 1 + -1i * 1) + i(3 + -1i)) + (2i + -1) = 0 ((3 + -1i) + i(3 + -1i)) + (2i + -1) = 0 (3 + -1i + (3 * i + -1i * i)) + (2i + -1) = 0 (3 + -1i + (3i + -1i2)) + (2i + -1) = 0 Combine like terms: -1i + 3i = 2i (3 + 2i + -1i2) + (2i + -1) = 0 Reorder the terms: 3 + 2i + -1i2 + (-1 + 2i) = 0 Remove parenthesis around (-1 + 2i) 3 + 2i + -1i2 + -1 + 2i = 0 Reorder the terms: 3 + -1 + 2i + 2i + -1i2 = 0 Combine like terms: 3 + -1 = 2 2 + 2i + 2i + -1i2 = 0 Combine like terms: 2i + 2i = 4i 2 + 4i + -1i2 = 0 Solving 2 + 4i + -1i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + -4i + i2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -4i + 2 + i2 = 0 + 2 Reorder the terms: -2 + 2 + -4i + i2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -4i + i2 = 0 + 2 -4i + i2 = 0 + 2 Combine like terms: 0 + 2 = 2 -4i + i2 = 2 The i term is -4i. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4i + 4 + i2 = 2 + 4 Reorder the terms: 4 + -4i + i2 = 2 + 4 Combine like terms: 2 + 4 = 6 4 + -4i + i2 = 6 Factor a perfect square on the left side: (i + -2)(i + -2) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (i + -2) equal to 2.449489743 and -2.449489743.Subproblem 1
i + -2 = 2.449489743 Simplifying i + -2 = 2.449489743 Reorder the terms: -2 + i = 2.449489743 Solving -2 + i = 2.449489743 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + i = 2.449489743 + 2 Combine like terms: -2 + 2 = 0 0 + i = 2.449489743 + 2 i = 2.449489743 + 2 Combine like terms: 2.449489743 + 2 = 4.449489743 i = 4.449489743 Simplifying i = 4.449489743Subproblem 2
i + -2 = -2.449489743 Simplifying i + -2 = -2.449489743 Reorder the terms: -2 + i = -2.449489743 Solving -2 + i = -2.449489743 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + i = -2.449489743 + 2 Combine like terms: -2 + 2 = 0 0 + i = -2.449489743 + 2 i = -2.449489743 + 2 Combine like terms: -2.449489743 + 2 = -0.449489743 i = -0.449489743 Simplifying i = -0.449489743Solution
The solution to the problem is based on the solutions from the subproblems. i = {4.449489743, -0.449489743}
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