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Simplifying 0 = (3 + 4i)(3 + 4i) Multiply (3 + 4i) * (3 + 4i) 0 = (3(3 + 4i) + 4i * (3 + 4i)) 0 = ((3 * 3 + 4i * 3) + 4i * (3 + 4i)) 0 = ((9 + 12i) + 4i * (3 + 4i)) 0 = (9 + 12i + (3 * 4i + 4i * 4i)) 0 = (9 + 12i + (12i + 16i2)) Combine like terms: 12i + 12i = 24i 0 = (9 + 24i + 16i2) Solving 0 = 9 + 24i + 16i2 Solving for variable 'i'. Combine like terms: 0 + -9 = -9 -9 + -24i + -16i2 = 9 + 24i + 16i2 + -9 + -24i + -16i2 Reorder the terms: -9 + -24i + -16i2 = 9 + -9 + 24i + -24i + 16i2 + -16i2 Combine like terms: 9 + -9 = 0 -9 + -24i + -16i2 = 0 + 24i + -24i + 16i2 + -16i2 -9 + -24i + -16i2 = 24i + -24i + 16i2 + -16i2 Combine like terms: 24i + -24i = 0 -9 + -24i + -16i2 = 0 + 16i2 + -16i2 -9 + -24i + -16i2 = 16i2 + -16i2 Combine like terms: 16i2 + -16i2 = 0 -9 + -24i + -16i2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(9 + 24i + 16i2) = 0 Factor a trinomial. -1((3 + 4i)(3 + 4i)) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(3 + 4i)' equal to zero and attempt to solve: Simplifying 3 + 4i = 0 Solving 3 + 4i = 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 + 4i = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 4i = 0 + -3 4i = 0 + -3 Combine like terms: 0 + -3 = -3 4i = -3 Divide each side by '4'. i = -0.75 Simplifying i = -0.75Subproblem 2
Set the factor '(3 + 4i)' equal to zero and attempt to solve: Simplifying 3 + 4i = 0 Solving 3 + 4i = 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 + 4i = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 4i = 0 + -3 4i = 0 + -3 Combine like terms: 0 + -3 = -3 4i = -3 Divide each side by '4'. i = -0.75 Simplifying i = -0.75Solution
i = {-0.75, -0.75}
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