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Simplifying (5 + i)(3 + -2i) = 4 Multiply (5 + i) * (3 + -2i) (5(3 + -2i) + i(3 + -2i)) = 4 ((3 * 5 + -2i * 5) + i(3 + -2i)) = 4 ((15 + -10i) + i(3 + -2i)) = 4 (15 + -10i + (3 * i + -2i * i)) = 4 (15 + -10i + (3i + -2i2)) = 4 Combine like terms: -10i + 3i = -7i (15 + -7i + -2i2) = 4 Solving 15 + -7i + -2i2 = 4 Solving for variable 'i'. Reorder the terms: 15 + -4 + -7i + -2i2 = 4 + -4 Combine like terms: 15 + -4 = 11 11 + -7i + -2i2 = 4 + -4 Combine like terms: 4 + -4 = 0 11 + -7i + -2i2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -5.5 + 3.5i + i2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + 3.5i + 5.5 + i2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + 3.5i + i2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + 3.5i + i2 = 0 + 5.5 3.5i + i2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 3.5i + i2 = 5.5 The i term is 3.5i. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5i + 3.0625 + i2 = 5.5 + 3.0625 Reorder the terms: 3.0625 + 3.5i + i2 = 5.5 + 3.0625 Combine like terms: 5.5 + 3.0625 = 8.5625 3.0625 + 3.5i + i2 = 8.5625 Factor a perfect square on the left side: (i + 1.75)(i + 1.75) = 8.5625 Calculate the square root of the right side: 2.926174978 Break this problem into two subproblems by setting (i + 1.75) equal to 2.926174978 and -2.926174978.Subproblem 1
i + 1.75 = 2.926174978 Simplifying i + 1.75 = 2.926174978 Reorder the terms: 1.75 + i = 2.926174978 Solving 1.75 + i = 2.926174978 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + i = 2.926174978 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + i = 2.926174978 + -1.75 i = 2.926174978 + -1.75 Combine like terms: 2.926174978 + -1.75 = 1.176174978 i = 1.176174978 Simplifying i = 1.176174978Subproblem 2
i + 1.75 = -2.926174978 Simplifying i + 1.75 = -2.926174978 Reorder the terms: 1.75 + i = -2.926174978 Solving 1.75 + i = -2.926174978 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + i = -2.926174978 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + i = -2.926174978 + -1.75 i = -2.926174978 + -1.75 Combine like terms: -2.926174978 + -1.75 = -4.676174978 i = -4.676174978 Simplifying i = -4.676174978Solution
The solution to the problem is based on the solutions from the subproblems. i = {1.176174978, -4.676174978}
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