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Simplifying (5i + 2)(4 + -3i) + -1(2 + -1i)(6i + -7) = 0 Reorder the terms: (2 + 5i)(4 + -3i) + -1(2 + -1i)(6i + -7) = 0 Multiply (2 + 5i) * (4 + -3i) (2(4 + -3i) + 5i * (4 + -3i)) + -1(2 + -1i)(6i + -7) = 0 ((4 * 2 + -3i * 2) + 5i * (4 + -3i)) + -1(2 + -1i)(6i + -7) = 0 ((8 + -6i) + 5i * (4 + -3i)) + -1(2 + -1i)(6i + -7) = 0 (8 + -6i + (4 * 5i + -3i * 5i)) + -1(2 + -1i)(6i + -7) = 0 (8 + -6i + (20i + -15i2)) + -1(2 + -1i)(6i + -7) = 0 Combine like terms: -6i + 20i = 14i (8 + 14i + -15i2) + -1(2 + -1i)(6i + -7) = 0 Reorder the terms: 8 + 14i + -15i2 + -1(2 + -1i)(-7 + 6i) = 0 Multiply (2 + -1i) * (-7 + 6i) 8 + 14i + -15i2 + -1(2(-7 + 6i) + -1i * (-7 + 6i)) = 0 8 + 14i + -15i2 + -1((-7 * 2 + 6i * 2) + -1i * (-7 + 6i)) = 0 8 + 14i + -15i2 + -1((-14 + 12i) + -1i * (-7 + 6i)) = 0 8 + 14i + -15i2 + -1(-14 + 12i + (-7 * -1i + 6i * -1i)) = 0 8 + 14i + -15i2 + -1(-14 + 12i + (7i + -6i2)) = 0 Combine like terms: 12i + 7i = 19i 8 + 14i + -15i2 + -1(-14 + 19i + -6i2) = 0 8 + 14i + -15i2 + (-14 * -1 + 19i * -1 + -6i2 * -1) = 0 8 + 14i + -15i2 + (14 + -19i + 6i2) = 0 Reorder the terms: 8 + 14 + 14i + -19i + -15i2 + 6i2 = 0 Combine like terms: 8 + 14 = 22 22 + 14i + -19i + -15i2 + 6i2 = 0 Combine like terms: 14i + -19i = -5i 22 + -5i + -15i2 + 6i2 = 0 Combine like terms: -15i2 + 6i2 = -9i2 22 + -5i + -9i2 = 0 Solving 22 + -5i + -9i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -2.444444444 + 0.5555555556i + i2 = 0 Move the constant term to the right: Add '2.444444444' to each side of the equation. -2.444444444 + 0.5555555556i + 2.444444444 + i2 = 0 + 2.444444444 Reorder the terms: -2.444444444 + 2.444444444 + 0.5555555556i + i2 = 0 + 2.444444444 Combine like terms: -2.444444444 + 2.444444444 = 0.000000000 0.000000000 + 0.5555555556i + i2 = 0 + 2.444444444 0.5555555556i + i2 = 0 + 2.444444444 Combine like terms: 0 + 2.444444444 = 2.444444444 0.5555555556i + i2 = 2.444444444 The i term is 0.5555555556i. Take half its coefficient (0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. 0.5555555556i + 0.07716049384 + i2 = 2.444444444 + 0.07716049384 Reorder the terms: 0.07716049384 + 0.5555555556i + i2 = 2.444444444 + 0.07716049384 Combine like terms: 2.444444444 + 0.07716049384 = 2.52160493784 0.07716049384 + 0.5555555556i + i2 = 2.52160493784 Factor a perfect square on the left side: (i + 0.2777777778)(i + 0.2777777778) = 2.52160493784 Calculate the square root of the right side: 1.587956214 Break this problem into two subproblems by setting (i + 0.2777777778) equal to 1.587956214 and -1.587956214.Subproblem 1
i + 0.2777777778 = 1.587956214 Simplifying i + 0.2777777778 = 1.587956214 Reorder the terms: 0.2777777778 + i = 1.587956214 Solving 0.2777777778 + i = 1.587956214 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + i = 1.587956214 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + i = 1.587956214 + -0.2777777778 i = 1.587956214 + -0.2777777778 Combine like terms: 1.587956214 + -0.2777777778 = 1.3101784362 i = 1.3101784362 Simplifying i = 1.3101784362Subproblem 2
i + 0.2777777778 = -1.587956214 Simplifying i + 0.2777777778 = -1.587956214 Reorder the terms: 0.2777777778 + i = -1.587956214 Solving 0.2777777778 + i = -1.587956214 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + i = -1.587956214 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + i = -1.587956214 + -0.2777777778 i = -1.587956214 + -0.2777777778 Combine like terms: -1.587956214 + -0.2777777778 = -1.8657339918 i = -1.8657339918 Simplifying i = -1.8657339918Solution
The solution to the problem is based on the solutions from the subproblems. i = {1.3101784362, -1.8657339918}
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