<p>Find the square root of <span>\(9 + 40i\)</span>.</p>
Step-by-Step Solution
Key Concept: To find √(a + bi), assume √(a + bi) = x + yi, square both sides to get a system of equations, then solve using (x² - y²) = a and 2xy = b simultaneously with x² + y² = √(a² + b²).
<p><strong>Step 1:</strong> Let √(9 + 40i) = x + yi where x, y ∈ ℝ</p><p><strong>Step 2:</strong> Square both sides: (x + yi)² = 9 + 40i<br>x² - y² + 2xyi = 9 + 40i</p><p><strong>Step 3:</strong> Equate real and imaginary parts:<br>x² - y² = 9 ... (1)<br>2xy = 40, so xy = 20 ... (2)</p><p><strong>Step 4:</strong> Find x² + y² using the modulus: |9 + 40i| = √(81 + 1600) = √1681 = 41<br>So x² + y² = 41 ... (3)</p><p><strong>Step 5:</strong> From equations (1) and (3):<br>Adding: 2x² = 50 → x² = 25 → x = ±5<br>Subtracting: 2y² = 32 → y² = 16 → y = ±4</p><p><strong>Step 6:</strong> From xy = 20, we need x and y to have the same sign:<br>If x = 5, then y = 4; if x = -5, then y = -4</p><p><strong>Step 7:</strong> Verify: (5 + 4i)² = 25 - 16 + 40i = 9 + 40i ✓<br>(-5 - 4i)² = 25 - 16 + 40i = 9 + 40i ✓</p><p>∴ Answer: <strong>(5 + 4i) or -(5 + 4i)</strong></p>
Correct Answer: (5 + 4i) or -(5 + 4i)