<p>The number of solutions of the equation \(z^2 + \bar{z} = 0\) is</p>
Step-by-Step Solution
Key Concept: Express z = x + iy and separate into real and imaginary parts. The equation z² + z̄ = 0 yields two independent equations in x and y that must be satisfied simultaneously.
<p><strong>Step 1:</strong> Let z = x + iy where x, y ∈ ℝ. Then z̄ = x - iy.</p><p><strong>Step 2:</strong> Calculate z² = (x + iy)² = x² - y² + 2ixy</p><p><strong>Step 3:</strong> Substitute into z² + z̄ = 0:<br/>(x² - y² + 2ixy) + (x - iy) = 0<br/>(x² - y² + x) + i(2xy - y) = 0</p><p><strong>Step 4:</strong> For this complex number to equal zero, both real and imaginary parts must be zero:<br/>Real part: x² - y² + x = 0 ... (1)<br/>Imaginary part: 2xy - y = 0 ... (2)</p><p><strong>Step 5:</strong> From equation (2): y(2x - 1) = 0<br/>So y = 0 or x = 1/2</p><p><strong>Step 6:</strong> Case 1: If y = 0, from (1): x² + x = 0 → x(x + 1) = 0 → x = 0 or x = -1<br/>This gives z = 0 and z = -1</p><p><strong>Step 7:</strong> Case 2: If x = 1/2, from (1): 1/4 - y² + 1/2 = 0 → y² = 3/4 → y = ±√3/2<br/>This gives z = 1/2 + i√3/2 and z = 1/2 - i√3/2</p><p><strong>Step 8:</strong> Verification shows all four solutions satisfy the original equation.</p><p>∴ Answer: <strong>D (4 solutions)</strong></p>
Correct Answer: D