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Simplifying 4(3 + 2i)(4 + -5i) = 18 + 2i Multiply (3 + 2i) * (4 + -5i) 4(3(4 + -5i) + 2i * (4 + -5i)) = 18 + 2i 4((4 * 3 + -5i * 3) + 2i * (4 + -5i)) = 18 + 2i 4((12 + -15i) + 2i * (4 + -5i)) = 18 + 2i 4(12 + -15i + (4 * 2i + -5i * 2i)) = 18 + 2i 4(12 + -15i + (8i + -10i2)) = 18 + 2i Combine like terms: -15i + 8i = -7i 4(12 + -7i + -10i2) = 18 + 2i (12 * 4 + -7i * 4 + -10i2 * 4) = 18 + 2i (48 + -28i + -40i2) = 18 + 2i Solving 48 + -28i + -40i2 = 18 + 2i Solving for variable 'i'. Reorder the terms: 48 + -18 + -28i + -2i + -40i2 = 18 + 2i + -18 + -2i Combine like terms: 48 + -18 = 30 30 + -28i + -2i + -40i2 = 18 + 2i + -18 + -2i Combine like terms: -28i + -2i = -30i 30 + -30i + -40i2 = 18 + 2i + -18 + -2i Reorder the terms: 30 + -30i + -40i2 = 18 + -18 + 2i + -2i Combine like terms: 18 + -18 = 0 30 + -30i + -40i2 = 0 + 2i + -2i 30 + -30i + -40i2 = 2i + -2i Combine like terms: 2i + -2i = 0 30 + -30i + -40i2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(3 + -3i + -4i2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(3 + -3i + -4i2)' equal to zero and attempt to solve: Simplifying 3 + -3i + -4i2 = 0 Solving 3 + -3i + -4i2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + 0.75i + i2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.75i + 0.75 + i2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.75i + i2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.75i + i2 = 0 + 0.75 0.75i + i2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.75i + i2 = 0.75 The i term is 0.75i. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75i + 0.140625 + i2 = 0.75 + 0.140625 Reorder the terms: 0.140625 + 0.75i + i2 = 0.75 + 0.140625 Combine like terms: 0.75 + 0.140625 = 0.890625 0.140625 + 0.75i + i2 = 0.890625 Factor a perfect square on the left side: (i + 0.375)(i + 0.375) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (i + 0.375) equal to 0.943729304 and -0.943729304.Subproblem 1
i + 0.375 = 0.943729304 Simplifying i + 0.375 = 0.943729304 Reorder the terms: 0.375 + i = 0.943729304 Solving 0.375 + i = 0.943729304 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + i = 0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + i = 0.943729304 + -0.375 i = 0.943729304 + -0.375 Combine like terms: 0.943729304 + -0.375 = 0.568729304 i = 0.568729304 Simplifying i = 0.568729304Subproblem 2
i + 0.375 = -0.943729304 Simplifying i + 0.375 = -0.943729304 Reorder the terms: 0.375 + i = -0.943729304 Solving 0.375 + i = -0.943729304 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + i = -0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + i = -0.943729304 + -0.375 i = -0.943729304 + -0.375 Combine like terms: -0.943729304 + -0.375 = -1.318729304 i = -1.318729304 Simplifying i = -1.318729304Solution
The solution to the problem is based on the solutions from the subproblems. i = {0.568729304, -1.318729304}Solution
i = {0.568729304, -1.318729304}
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