<p>Solve \(\sqrt{x+5}+\sqrt{x+21}=\sqrt{6x+40}\).</p>
Step-by-Step Solution
Key Concept: Square both sides strategically to eliminate radicals, then recognize that the middle term simplifies when the cross-product equals one of the original radicals. Verify solutions in the original equation since squaring can introduce extraneous roots.
<p><strong>Step 1:</strong> Square both sides of √(x+5) + √(x+21) = √(6x+40)</p><p>(x+5) + 2√(x+5)√(x+21) + (x+21) = 6x+40</p><p><strong>Step 2:</strong> Simplify the left side:</p><p>2x + 26 + 2√((x+5)(x+21)) = 6x + 40</p><p><strong>Step 3:</strong> Isolate the radical term:</p><p>2√((x+5)(x+21)) = 4x + 14</p><p>√((x+5)(x+21)) = 2x + 7</p><p><strong>Step 4:</strong> Square again:</p><p>(x+5)(x+21) = (2x+7)²</p><p>x² + 26x + 105 = 4x² + 28x + 49</p><p><strong>Step 5:</strong> Rearrange:</p><p>0 = 3x² + 2x - 56</p><p>0 = (3x + 14)(x - 4)</p><p>x = -14/3 or x = 4</p><p><strong>Step 6:</strong> Check x = 4 in original equation:</p><p>√9 + √25 = √64 → 3 + 5 = 8 ✓</p><p>Check x = -14/3: √(1/3) + √(49/3) ≠ √(-4/3) (undefined) ✗</p><p><strong>∴ Answer: x = 4</strong></p>
Correct Answer: x = 4