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Simplifying i(3 + 2i) + (8 + -3i)(1 + i) = 0 (3 * i + 2i * i) + (8 + -3i)(1 + i) = 0 (3i + 2i2) + (8 + -3i)(1 + i) = 0 Multiply (8 + -3i) * (1 + i) 3i + 2i2 + (8(1 + i) + -3i * (1 + i)) = 0 3i + 2i2 + ((1 * 8 + i * 8) + -3i * (1 + i)) = 0 3i + 2i2 + ((8 + 8i) + -3i * (1 + i)) = 0 3i + 2i2 + (8 + 8i + (1 * -3i + i * -3i)) = 0 3i + 2i2 + (8 + 8i + (-3i + -3i2)) = 0 Combine like terms: 8i + -3i = 5i 3i + 2i2 + (8 + 5i + -3i2) = 0 Reorder the terms: 8 + 3i + 5i + 2i2 + -3i2 = 0 Combine like terms: 3i + 5i = 8i 8 + 8i + 2i2 + -3i2 = 0 Combine like terms: 2i2 + -3i2 = -1i2 8 + 8i + -1i2 = 0 Solving 8 + 8i + -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'. -8 + -8i + i2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -8i + 8 + i2 = 0 + 8 Reorder the terms: -8 + 8 + -8i + i2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -8i + i2 = 0 + 8 -8i + i2 = 0 + 8 Combine like terms: 0 + 8 = 8 -8i + i2 = 8 The i term is -8i. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8i + 16 + i2 = 8 + 16 Reorder the terms: 16 + -8i + i2 = 8 + 16 Combine like terms: 8 + 16 = 24 16 + -8i + i2 = 24 Factor a perfect square on the left side: (i + -4)(i + -4) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (i + -4) equal to 4.898979486 and -4.898979486.Subproblem 1
i + -4 = 4.898979486 Simplifying i + -4 = 4.898979486 Reorder the terms: -4 + i = 4.898979486 Solving -4 + i = 4.898979486 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + i = 4.898979486 + 4 Combine like terms: -4 + 4 = 0 0 + i = 4.898979486 + 4 i = 4.898979486 + 4 Combine like terms: 4.898979486 + 4 = 8.898979486 i = 8.898979486 Simplifying i = 8.898979486Subproblem 2
i + -4 = -4.898979486 Simplifying i + -4 = -4.898979486 Reorder the terms: -4 + i = -4.898979486 Solving -4 + i = -4.898979486 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + i = -4.898979486 + 4 Combine like terms: -4 + 4 = 0 0 + i = -4.898979486 + 4 i = -4.898979486 + 4 Combine like terms: -4.898979486 + 4 = -0.898979486 i = -0.898979486 Simplifying i = -0.898979486Solution
The solution to the problem is based on the solutions from the subproblems. i = {8.898979486, -0.898979486}
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