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Simplifying 2 + i + (7 + i)(5 + -1i) = 0 Multiply (7 + i) * (5 + -1i) 2 + i + (7(5 + -1i) + i(5 + -1i)) = 0 2 + i + ((5 * 7 + -1i * 7) + i(5 + -1i)) = 0 2 + i + ((35 + -7i) + i(5 + -1i)) = 0 2 + i + (35 + -7i + (5 * i + -1i * i)) = 0 2 + i + (35 + -7i + (5i + -1i2)) = 0 Combine like terms: -7i + 5i = -2i 2 + i + (35 + -2i + -1i2) = 0 Reorder the terms: 2 + 35 + i + -2i + -1i2 = 0 Combine like terms: 2 + 35 = 37 37 + i + -2i + -1i2 = 0 Combine like terms: i + -2i = -1i 37 + -1i + -1i2 = 0 Solving 37 + -1i + -1i2 = 0 Solving for variable 'i'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -37 + i + i2 = 0 Move the constant term to the right: Add '37' to each side of the equation. -37 + i + 37 + i2 = 0 + 37 Reorder the terms: -37 + 37 + i + i2 = 0 + 37 Combine like terms: -37 + 37 = 0 0 + i + i2 = 0 + 37 i + i2 = 0 + 37 Combine like terms: 0 + 37 = 37 i + i2 = 37 The i term is i. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + i2 = 37 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + i2 = 37 + 0.25 Combine like terms: 37 + 0.25 = 37.25 1.25 + i2 = 37.25 Factor a perfect square on the left side: (i + 0.5)(i + 0.5) = 37.25 Calculate the square root of the right side: 6.103277808 Break this problem into two subproblems by setting (i + 0.5) equal to 6.103277808 and -6.103277808.Subproblem 1
i + 0.5 = 6.103277808 Simplifying i + 0.5 = 6.103277808 Reorder the terms: 0.5 + i = 6.103277808 Solving 0.5 + i = 6.103277808 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + i = 6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + i = 6.103277808 + -0.5 i = 6.103277808 + -0.5 Combine like terms: 6.103277808 + -0.5 = 5.603277808 i = 5.603277808 Simplifying i = 5.603277808Subproblem 2
i + 0.5 = -6.103277808 Simplifying i + 0.5 = -6.103277808 Reorder the terms: 0.5 + i = -6.103277808 Solving 0.5 + i = -6.103277808 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + i = -6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + i = -6.103277808 + -0.5 i = -6.103277808 + -0.5 Combine like terms: -6.103277808 + -0.5 = -6.603277808 i = -6.603277808 Simplifying i = -6.603277808Solution
The solution to the problem is based on the solutions from the subproblems. i = {5.603277808, -6.603277808}
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