Step-by-Step Solution
Key Concept: Since |z| is a non-negative real number, the right side z + 1 + 2i must also be real and non-negative. This means the imaginary part of z must equal -2, and we can then solve for the real part using the modulus definition.
<p><strong>Step 1:</strong> Let z = x + iy. Then |z| = √(x² + y²), which is a non-negative real number.</p><p><strong>Step 2:</strong> The equation becomes: √(x² + y²) = (x + 1) + i(y + 2)</p><p><strong>Step 3:</strong> Since the left side is real, the right side must be real. Therefore, the imaginary part must be zero: y + 2 = 0, so <strong>y = -2</strong></p><p><strong>Step 4:</strong> The equation reduces to: √(x² + 4) = x + 1</p><p><strong>Step 5:</strong> Square both sides: x² + 4 = x² + 2x + 1</p><p><strong>Step 6:</strong> Simplify: 4 = 2x + 1 → 2x = 3 → <strong>x = 3/2</strong></p><p><strong>Step 7:</strong> Verify: |3/2 - 2i| = √(9/4 + 4) = √(25/4) = 5/2, and (3/2 - 2i) + 1 + 2i = 5/2 ✓</p><p>∴ Answer: z = 3/2 - 2i</p>
Correct Answer: x + iy = 3/2 - 2i