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Simplifying (6 + -3.25i)(4.1 + 14i) = 0 Multiply (6 + -3.25i) * (4.1 + 14i) (6(4.1 + 14i) + -3.25i * (4.1 + 14i)) = 0 ((4.1 * 6 + 14i * 6) + -3.25i * (4.1 + 14i)) = 0 ((24.6 + 84i) + -3.25i * (4.1 + 14i)) = 0 (24.6 + 84i + (4.1 * -3.25i + 14i * -3.25i)) = 0 (24.6 + 84i + (-13.325i + -45.5i2)) = 0 Combine like terms: 84i + -13.325i = 70.675i (24.6 + 70.675i + -45.5i2) = 0 Solving 24.6 + 70.675i + -45.5i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -45.5 the coefficient of the squared term: Divide each side by '-45.5'. -0.5406593407 + -1.553296703i + i2 = 0 Move the constant term to the right: Add '0.5406593407' to each side of the equation. -0.5406593407 + -1.553296703i + 0.5406593407 + i2 = 0 + 0.5406593407 Reorder the terms: -0.5406593407 + 0.5406593407 + -1.553296703i + i2 = 0 + 0.5406593407 Combine like terms: -0.5406593407 + 0.5406593407 = 0.0000000000 0.0000000000 + -1.553296703i + i2 = 0 + 0.5406593407 -1.553296703i + i2 = 0 + 0.5406593407 Combine like terms: 0 + 0.5406593407 = 0.5406593407 -1.553296703i + i2 = 0.5406593407 The i term is -1.553296703i. Take half its coefficient (-0.7766483515). Square it (0.6031826619) and add it to both sides. Add '0.6031826619' to each side of the equation. -1.553296703i + 0.6031826619 + i2 = 0.5406593407 + 0.6031826619 Reorder the terms: 0.6031826619 + -1.553296703i + i2 = 0.5406593407 + 0.6031826619 Combine like terms: 0.5406593407 + 0.6031826619 = 1.1438420026 0.6031826619 + -1.553296703i + i2 = 1.1438420026 Factor a perfect square on the left side: (i + -0.7766483515)(i + -0.7766483515) = 1.1438420026 Calculate the square root of the right side: 1.069505494 Break this problem into two subproblems by setting (i + -0.7766483515) equal to 1.069505494 and -1.069505494.Subproblem 1
i + -0.7766483515 = 1.069505494 Simplifying i + -0.7766483515 = 1.069505494 Reorder the terms: -0.7766483515 + i = 1.069505494 Solving -0.7766483515 + i = 1.069505494 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.7766483515' to each side of the equation. -0.7766483515 + 0.7766483515 + i = 1.069505494 + 0.7766483515 Combine like terms: -0.7766483515 + 0.7766483515 = 0.0000000000 0.0000000000 + i = 1.069505494 + 0.7766483515 i = 1.069505494 + 0.7766483515 Combine like terms: 1.069505494 + 0.7766483515 = 1.8461538455 i = 1.8461538455 Simplifying i = 1.8461538455Subproblem 2
i + -0.7766483515 = -1.069505494 Simplifying i + -0.7766483515 = -1.069505494 Reorder the terms: -0.7766483515 + i = -1.069505494 Solving -0.7766483515 + i = -1.069505494 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.7766483515' to each side of the equation. -0.7766483515 + 0.7766483515 + i = -1.069505494 + 0.7766483515 Combine like terms: -0.7766483515 + 0.7766483515 = 0.0000000000 0.0000000000 + i = -1.069505494 + 0.7766483515 i = -1.069505494 + 0.7766483515 Combine like terms: -1.069505494 + 0.7766483515 = -0.2928571425 i = -0.2928571425 Simplifying i = -0.2928571425Solution
The solution to the problem is based on the solutions from the subproblems. i = {1.8461538455, -0.2928571425}
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