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Simplifying 0 = (3 + 2i)(4 + i) Multiply (3 + 2i) * (4 + i) 0 = (3(4 + i) + 2i * (4 + i)) 0 = ((4 * 3 + i * 3) + 2i * (4 + i)) 0 = ((12 + 3i) + 2i * (4 + i)) 0 = (12 + 3i + (4 * 2i + i * 2i)) 0 = (12 + 3i + (8i + 2i2)) Combine like terms: 3i + 8i = 11i 0 = (12 + 11i + 2i2) Solving 0 = 12 + 11i + 2i2 Solving for variable 'i'. Combine like terms: 0 + -12 = -12 -12 + -11i + -2i2 = 12 + 11i + 2i2 + -12 + -11i + -2i2 Reorder the terms: -12 + -11i + -2i2 = 12 + -12 + 11i + -11i + 2i2 + -2i2 Combine like terms: 12 + -12 = 0 -12 + -11i + -2i2 = 0 + 11i + -11i + 2i2 + -2i2 -12 + -11i + -2i2 = 11i + -11i + 2i2 + -2i2 Combine like terms: 11i + -11i = 0 -12 + -11i + -2i2 = 0 + 2i2 + -2i2 -12 + -11i + -2i2 = 2i2 + -2i2 Combine like terms: 2i2 + -2i2 = 0 -12 + -11i + -2i2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(12 + 11i + 2i2) = 0 Factor a trinomial. -1((4 + i)(3 + 2i)) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(4 + i)' equal to zero and attempt to solve: Simplifying 4 + i = 0 Solving 4 + i = 0 Move all terms containing i to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + i = 0 + -4 Combine like terms: 4 + -4 = 0 0 + i = 0 + -4 i = 0 + -4 Combine like terms: 0 + -4 = -4 i = -4 Simplifying i = -4Subproblem 2
Set the factor '(3 + 2i)' equal to zero and attempt to solve: Simplifying 3 + 2i = 0 Solving 3 + 2i = 0 Move all terms containing i to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + 2i = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 2i = 0 + -3 2i = 0 + -3 Combine like terms: 0 + -3 = -3 2i = -3 Divide each side by '2'. i = -1.5 Simplifying i = -1.5Solution
i = {-4, -1.5}
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