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Simplifying 7 + -8i + 6i(3 + -4i) = 0 7 + -8i + (3 * 6i + -4i * 6i) = 0 7 + -8i + (18i + -24i2) = 0 Combine like terms: -8i + 18i = 10i 7 + 10i + -24i2 = 0 Solving 7 + 10i + -24i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.2916666667 + -0.4166666667i + i2 = 0 Move the constant term to the right: Add '0.2916666667' to each side of the equation. -0.2916666667 + -0.4166666667i + 0.2916666667 + i2 = 0 + 0.2916666667 Reorder the terms: -0.2916666667 + 0.2916666667 + -0.4166666667i + i2 = 0 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + -0.4166666667i + i2 = 0 + 0.2916666667 -0.4166666667i + i2 = 0 + 0.2916666667 Combine like terms: 0 + 0.2916666667 = 0.2916666667 -0.4166666667i + i2 = 0.2916666667 The i term is -0.4166666667i. Take half its coefficient (-0.2083333334). Square it (0.04340277781) and add it to both sides. Add '0.04340277781' to each side of the equation. -0.4166666667i + 0.04340277781 + i2 = 0.2916666667 + 0.04340277781 Reorder the terms: 0.04340277781 + -0.4166666667i + i2 = 0.2916666667 + 0.04340277781 Combine like terms: 0.2916666667 + 0.04340277781 = 0.33506944451 0.04340277781 + -0.4166666667i + i2 = 0.33506944451 Factor a perfect square on the left side: (i + -0.2083333334)(i + -0.2083333334) = 0.33506944451 Calculate the square root of the right side: 0.578851833 Break this problem into two subproblems by setting (i + -0.2083333334) equal to 0.578851833 and -0.578851833.Subproblem 1
i + -0.2083333334 = 0.578851833 Simplifying i + -0.2083333334 = 0.578851833 Reorder the terms: -0.2083333334 + i = 0.578851833 Solving -0.2083333334 + i = 0.578851833 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + i = 0.578851833 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + i = 0.578851833 + 0.2083333334 i = 0.578851833 + 0.2083333334 Combine like terms: 0.578851833 + 0.2083333334 = 0.7871851664 i = 0.7871851664 Simplifying i = 0.7871851664Subproblem 2
i + -0.2083333334 = -0.578851833 Simplifying i + -0.2083333334 = -0.578851833 Reorder the terms: -0.2083333334 + i = -0.578851833 Solving -0.2083333334 + i = -0.578851833 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.2083333334' to each side of the equation. -0.2083333334 + 0.2083333334 + i = -0.578851833 + 0.2083333334 Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000 0.0000000000 + i = -0.578851833 + 0.2083333334 i = -0.578851833 + 0.2083333334 Combine like terms: -0.578851833 + 0.2083333334 = -0.3705184996 i = -0.3705184996 Simplifying i = -0.3705184996Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.7871851664, -0.3705184996}
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