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Simplifying (2 + 7i) * i + -1(3 + 5i) = 0 Reorder the terms for easier multiplication: i(2 + 7i) + -1(3 + 5i) = 0 (2 * i + 7i * i) + -1(3 + 5i) = 0 (2i + 7i2) + -1(3 + 5i) = 0 2i + 7i2 + (3 * -1 + 5i * -1) = 0 2i + 7i2 + (-3 + -5i) = 0 Reorder the terms: -3 + 2i + -5i + 7i2 = 0 Combine like terms: 2i + -5i = -3i -3 + -3i + 7i2 = 0 Solving -3 + -3i + 7i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + -0.4285714286i + i2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + -0.4285714286i + 0.4285714286 + i2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + -0.4285714286i + i2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + -0.4285714286i + i2 = 0 + 0.4285714286 -0.4285714286i + i2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 -0.4285714286i + i2 = 0.4285714286 The i term is -0.4285714286i. Take half its coefficient (-0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. -0.4285714286i + 0.04591836735 + i2 = 0.4285714286 + 0.04591836735 Reorder the terms: 0.04591836735 + -0.4285714286i + i2 = 0.4285714286 + 0.04591836735 Combine like terms: 0.4285714286 + 0.04591836735 = 0.47448979595 0.04591836735 + -0.4285714286i + i2 = 0.47448979595 Factor a perfect square on the left side: (i + -0.2142857143)(i + -0.2142857143) = 0.47448979595 Calculate the square root of the right side: 0.688832197 Break this problem into two subproblems by setting (i + -0.2142857143) equal to 0.688832197 and -0.688832197.Subproblem 1
i + -0.2142857143 = 0.688832197 Simplifying i + -0.2142857143 = 0.688832197 Reorder the terms: -0.2142857143 + i = 0.688832197 Solving -0.2142857143 + i = 0.688832197 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + i = 0.688832197 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + i = 0.688832197 + 0.2142857143 i = 0.688832197 + 0.2142857143 Combine like terms: 0.688832197 + 0.2142857143 = 0.9031179113 i = 0.9031179113 Simplifying i = 0.9031179113Subproblem 2
i + -0.2142857143 = -0.688832197 Simplifying i + -0.2142857143 = -0.688832197 Reorder the terms: -0.2142857143 + i = -0.688832197 Solving -0.2142857143 + i = -0.688832197 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.2142857143' to each side of the equation. -0.2142857143 + 0.2142857143 + i = -0.688832197 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + i = -0.688832197 + 0.2142857143 i = -0.688832197 + 0.2142857143 Combine like terms: -0.688832197 + 0.2142857143 = -0.4745464827 i = -0.4745464827 Simplifying i = -0.4745464827Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.9031179113, -0.4745464827}
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