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Simplifying (2 + 7i) * i + -1(3 + 5i)(5 + i) = 0 Reorder the terms for easier multiplication: i(2 + 7i) + -1(3 + 5i)(5 + i) = 0 (2 * i + 7i * i) + -1(3 + 5i)(5 + i) = 0 (2i + 7i2) + -1(3 + 5i)(5 + i) = 0 Multiply (3 + 5i) * (5 + i) 2i + 7i2 + -1(3(5 + i) + 5i * (5 + i)) = 0 2i + 7i2 + -1((5 * 3 + i * 3) + 5i * (5 + i)) = 0 2i + 7i2 + -1((15 + 3i) + 5i * (5 + i)) = 0 2i + 7i2 + -1(15 + 3i + (5 * 5i + i * 5i)) = 0 2i + 7i2 + -1(15 + 3i + (25i + 5i2)) = 0 Combine like terms: 3i + 25i = 28i 2i + 7i2 + -1(15 + 28i + 5i2) = 0 2i + 7i2 + (15 * -1 + 28i * -1 + 5i2 * -1) = 0 2i + 7i2 + (-15 + -28i + -5i2) = 0 Reorder the terms: -15 + 2i + -28i + 7i2 + -5i2 = 0 Combine like terms: 2i + -28i = -26i -15 + -26i + 7i2 + -5i2 = 0 Combine like terms: 7i2 + -5i2 = 2i2 -15 + -26i + 2i2 = 0 Solving -15 + -26i + 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'. -7.5 + -13i + i2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -13i + 7.5 + i2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -13i + i2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -13i + i2 = 0 + 7.5 -13i + i2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -13i + i2 = 7.5 The i term is -13i. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13i + 42.25 + i2 = 7.5 + 42.25 Reorder the terms: 42.25 + -13i + i2 = 7.5 + 42.25 Combine like terms: 7.5 + 42.25 = 49.75 42.25 + -13i + i2 = 49.75 Factor a perfect square on the left side: (i + -6.5)(i + -6.5) = 49.75 Calculate the square root of the right side: 7.05336799 Break this problem into two subproblems by setting (i + -6.5) equal to 7.05336799 and -7.05336799.Subproblem 1
i + -6.5 = 7.05336799 Simplifying i + -6.5 = 7.05336799 Reorder the terms: -6.5 + i = 7.05336799 Solving -6.5 + i = 7.05336799 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + i = 7.05336799 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + i = 7.05336799 + 6.5 i = 7.05336799 + 6.5 Combine like terms: 7.05336799 + 6.5 = 13.55336799 i = 13.55336799 Simplifying i = 13.55336799Subproblem 2
i + -6.5 = -7.05336799 Simplifying i + -6.5 = -7.05336799 Reorder the terms: -6.5 + i = -7.05336799 Solving -6.5 + i = -7.05336799 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + i = -7.05336799 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + i = -7.05336799 + 6.5 i = -7.05336799 + 6.5 Combine like terms: -7.05336799 + 6.5 = -0.55336799 i = -0.55336799 Simplifying i = -0.55336799Solution
The solution to the problem is based on the solutions from the subproblems. i = {13.55336799, -0.55336799}
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