Step 1: Group real and imaginary parts: (3 + 2i) + (5 - 4i) = (3 + 5) + (2i - 4i)

Step 2: Combine like terms: Real parts: 3 + 5 = 8 Imaginary parts: 2i - 4i = -2i

Step 3: Write in standard form: 8 - 2i

Answer: 8 - 2i

Combine real parts and imaginary parts separately:

Real parts: $5 + (-3) = 2$ Imaginary parts: $-2i + 7i = 5i$

Answer:

Step 1: Use FOIL method: First: 2 × 4 = 8 Outer: 2 × (-i) = -2i Inner: 3i × 4 = 12i Last: 3i × (-i) = -3i²

Step 2: Combine: 8 - 2i + 12i - 3i²

Step 3: Remember that i² = -1: -3i² = -3(-1) = 3

Step 4: Combine all terms: 8 - 2i + 12i + 3 = (8 + 3) + (-2i + 12i) = 11 + 10i

Answer: 11 + 10i

Step 1: Multiply by conjugate of denominator: The conjugate of (1 - i) is (1 + i)

Step 2: Multiply numerator and denominator: (6 + 8i)/(1 - i) × (1 + i)/(1 + i)

Step 3: Expand numerator: (6 + 8i)(1 + i) = 6 + 6i + 8i + 8i² = 6 + 14i + 8(-1) = 6 + 14i - 8 = -2 + 14i

Step 4: Expand denominator: (1 - i)(1 + i) = 1 + i - i - i² = 1 - (-1) = 1 + 1 = 2

Step 5: Divide: (-2 + 14i)/2 = -2/2 + 14i/2 = -1 + 7i

Answer: -1 + 7i

Notice these are conjugates! Use the formula $(a + bi)(a - bi) = a^2 + b^2$

Or use FOIL: $(3 - 2i)(3 + 2i)$ $= 9 + 6i - 6i - 4i^2$

Answer: $13$ (a real number!)

Step 1: Solve for x²: x² = -4

Step 2: Take square root of both sides: x = ±√(-4)

Step 3: Simplify √(-4): √(-4) = √(4 × (-1)) = √4 × √(-1) = 2i

Step 4: Write both solutions: x = 2i or x = -2i

Step 5: Verify x = 2i: (2i)² + 4 = 4i² + 4 = 4(-1) + 4 = -4 + 4 = 0 ✓

Step 6: Verify x = -2i: (-2i)² + 4 = 4i² + 4 = 4(-1) + 4 = -4 + 4 = 0 ✓

Answer: x = ±2i