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Simplifying (5 + -4i)(0.1 + -1i) = 0 Multiply (5 + -4i) * (0.1 + -1i) (5(0.1 + -1i) + -4i * (0.1 + -1i)) = 0 ((0.1 * 5 + -1i * 5) + -4i * (0.1 + -1i)) = 0 ((0.5 + -5i) + -4i * (0.1 + -1i)) = 0 (0.5 + -5i + (0.1 * -4i + -1i * -4i)) = 0 (0.5 + -5i + (-0.4i + 4i2)) = 0 Combine like terms: -5i + -0.4i = -5.4i (0.5 + -5.4i + 4i2) = 0 Solving 0.5 + -5.4i + 4i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.125 + -1.35i + i2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1.35i + -0.125 + i2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1.35i + i2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1.35i + i2 = 0 + -0.125 -1.35i + i2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1.35i + i2 = -0.125 The i term is -1.35i. Take half its coefficient (-0.675). Square it (0.455625) and add it to both sides. Add '0.455625' to each side of the equation. -1.35i + 0.455625 + i2 = -0.125 + 0.455625 Reorder the terms: 0.455625 + -1.35i + i2 = -0.125 + 0.455625 Combine like terms: -0.125 + 0.455625 = 0.330625 0.455625 + -1.35i + i2 = 0.330625 Factor a perfect square on the left side: (i + -0.675)(i + -0.675) = 0.330625 Calculate the square root of the right side: 0.575 Break this problem into two subproblems by setting (i + -0.675) equal to 0.575 and -0.575.Subproblem 1
i + -0.675 = 0.575 Simplifying i + -0.675 = 0.575 Reorder the terms: -0.675 + i = 0.575 Solving -0.675 + i = 0.575 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.675' to each side of the equation. -0.675 + 0.675 + i = 0.575 + 0.675 Combine like terms: -0.675 + 0.675 = 0.000 0.000 + i = 0.575 + 0.675 i = 0.575 + 0.675 Combine like terms: 0.575 + 0.675 = 1.25 i = 1.25 Simplifying i = 1.25Subproblem 2
i + -0.675 = -0.575 Simplifying i + -0.675 = -0.575 Reorder the terms: -0.675 + i = -0.575 Solving -0.675 + i = -0.575 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.675' to each side of the equation. -0.675 + 0.675 + i = -0.575 + 0.675 Combine like terms: -0.675 + 0.675 = 0.000 0.000 + i = -0.575 + 0.675 i = -0.575 + 0.675 Combine like terms: -0.575 + 0.675 = 0.1 i = 0.1 Simplifying i = 0.1Solution
The solution to the problem is based on the solutions from the subproblems. i = {1.25, 0.1}
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