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Simplifying (3 + 5i)(7 + -2i) + (-2 + 6i)(3 + -2i) = 0 Multiply (3 + 5i) * (7 + -2i) (3(7 + -2i) + 5i * (7 + -2i)) + (-2 + 6i)(3 + -2i) = 0 ((7 * 3 + -2i * 3) + 5i * (7 + -2i)) + (-2 + 6i)(3 + -2i) = 0 ((21 + -6i) + 5i * (7 + -2i)) + (-2 + 6i)(3 + -2i) = 0 (21 + -6i + (7 * 5i + -2i * 5i)) + (-2 + 6i)(3 + -2i) = 0 (21 + -6i + (35i + -10i2)) + (-2 + 6i)(3 + -2i) = 0 Combine like terms: -6i + 35i = 29i (21 + 29i + -10i2) + (-2 + 6i)(3 + -2i) = 0 Multiply (-2 + 6i) * (3 + -2i) 21 + 29i + -10i2 + (-2(3 + -2i) + 6i * (3 + -2i)) = 0 21 + 29i + -10i2 + ((3 * -2 + -2i * -2) + 6i * (3 + -2i)) = 0 21 + 29i + -10i2 + ((-6 + 4i) + 6i * (3 + -2i)) = 0 21 + 29i + -10i2 + (-6 + 4i + (3 * 6i + -2i * 6i)) = 0 21 + 29i + -10i2 + (-6 + 4i + (18i + -12i2)) = 0 Combine like terms: 4i + 18i = 22i 21 + 29i + -10i2 + (-6 + 22i + -12i2) = 0 Reorder the terms: 21 + -6 + 29i + 22i + -10i2 + -12i2 = 0 Combine like terms: 21 + -6 = 15 15 + 29i + 22i + -10i2 + -12i2 = 0 Combine like terms: 29i + 22i = 51i 15 + 51i + -10i2 + -12i2 = 0 Combine like terms: -10i2 + -12i2 = -22i2 15 + 51i + -22i2 = 0 Solving 15 + 51i + -22i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -22 the coefficient of the squared term: Divide each side by '-22'. -0.6818181818 + -2.318181818i + i2 = 0 Move the constant term to the right: Add '0.6818181818' to each side of the equation. -0.6818181818 + -2.318181818i + 0.6818181818 + i2 = 0 + 0.6818181818 Reorder the terms: -0.6818181818 + 0.6818181818 + -2.318181818i + i2 = 0 + 0.6818181818 Combine like terms: -0.6818181818 + 0.6818181818 = 0.0000000000 0.0000000000 + -2.318181818i + i2 = 0 + 0.6818181818 -2.318181818i + i2 = 0 + 0.6818181818 Combine like terms: 0 + 0.6818181818 = 0.6818181818 -2.318181818i + i2 = 0.6818181818 The i term is -2.318181818i. Take half its coefficient (-1.159090909). Square it (1.343491735) and add it to both sides. Add '1.343491735' to each side of the equation. -2.318181818i + 1.343491735 + i2 = 0.6818181818 + 1.343491735 Reorder the terms: 1.343491735 + -2.318181818i + i2 = 0.6818181818 + 1.343491735 Combine like terms: 0.6818181818 + 1.343491735 = 2.0253099168 1.343491735 + -2.318181818i + i2 = 2.0253099168 Factor a perfect square on the left side: (i + -1.159090909)(i + -1.159090909) = 2.0253099168 Calculate the square root of the right side: 1.423133837 Break this problem into two subproblems by setting (i + -1.159090909) equal to 1.423133837 and -1.423133837.Subproblem 1
i + -1.159090909 = 1.423133837 Simplifying i + -1.159090909 = 1.423133837 Reorder the terms: -1.159090909 + i = 1.423133837 Solving -1.159090909 + i = 1.423133837 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '1.159090909' to each side of the equation. -1.159090909 + 1.159090909 + i = 1.423133837 + 1.159090909 Combine like terms: -1.159090909 + 1.159090909 = 0.000000000 0.000000000 + i = 1.423133837 + 1.159090909 i = 1.423133837 + 1.159090909 Combine like terms: 1.423133837 + 1.159090909 = 2.582224746 i = 2.582224746 Simplifying i = 2.582224746Subproblem 2
i + -1.159090909 = -1.423133837 Simplifying i + -1.159090909 = -1.423133837 Reorder the terms: -1.159090909 + i = -1.423133837 Solving -1.159090909 + i = -1.423133837 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '1.159090909' to each side of the equation. -1.159090909 + 1.159090909 + i = -1.423133837 + 1.159090909 Combine like terms: -1.159090909 + 1.159090909 = 0.000000000 0.000000000 + i = -1.423133837 + 1.159090909 i = -1.423133837 + 1.159090909 Combine like terms: -1.423133837 + 1.159090909 = -0.264042928 i = -0.264042928 Simplifying i = -0.264042928Solution
The solution to the problem is based on the solutions from the subproblems. i = {2.582224746, -0.264042928}
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