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Simplifying i(4 + -1i) + 3(i + -3) = 0 (4 * i + -1i * i) + 3(i + -3) = 0 (4i + -1i2) + 3(i + -3) = 0 Reorder the terms: 4i + -1i2 + 3(-3 + i) = 0 4i + -1i2 + (-3 * 3 + i * 3) = 0 4i + -1i2 + (-9 + 3i) = 0 Reorder the terms: -9 + 4i + 3i + -1i2 = 0 Combine like terms: 4i + 3i = 7i -9 + 7i + -1i2 = 0 Solving -9 + 7i + -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'. 9 + -7i + i2 = 0 Move the constant term to the right: Add '-9' to each side of the equation. 9 + -7i + -9 + i2 = 0 + -9 Reorder the terms: 9 + -9 + -7i + i2 = 0 + -9 Combine like terms: 9 + -9 = 0 0 + -7i + i2 = 0 + -9 -7i + i2 = 0 + -9 Combine like terms: 0 + -9 = -9 -7i + i2 = -9 The i term is -7i. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7i + 12.25 + i2 = -9 + 12.25 Reorder the terms: 12.25 + -7i + i2 = -9 + 12.25 Combine like terms: -9 + 12.25 = 3.25 12.25 + -7i + i2 = 3.25 Factor a perfect square on the left side: (i + -3.5)(i + -3.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (i + -3.5) equal to 1.802775638 and -1.802775638.Subproblem 1
i + -3.5 = 1.802775638 Simplifying i + -3.5 = 1.802775638 Reorder the terms: -3.5 + i = 1.802775638 Solving -3.5 + i = 1.802775638 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + i = 1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + i = 1.802775638 + 3.5 i = 1.802775638 + 3.5 Combine like terms: 1.802775638 + 3.5 = 5.302775638 i = 5.302775638 Simplifying i = 5.302775638Subproblem 2
i + -3.5 = -1.802775638 Simplifying i + -3.5 = -1.802775638 Reorder the terms: -3.5 + i = -1.802775638 Solving -3.5 + i = -1.802775638 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + i = -1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + i = -1.802775638 + 3.5 i = -1.802775638 + 3.5 Combine like terms: -1.802775638 + 3.5 = 1.697224362 i = 1.697224362 Simplifying i = 1.697224362Solution
The solution to the problem is based on the solutions from the subproblems. i = {5.302775638, 1.697224362}
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