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