<p>Solve \(\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1\).</p>
Step-by-Step Solution
Key Concept: Use substitution u = 5x² - 6x to transform the radical equation into a simpler form, then rationalize by multiplying by the conjugate to eliminate nested radicals.
<p><strong>Step 1:</strong> Let u = 5x² - 6x. The equation becomes √(u + 8) - √(u - 7) = 1.</p><p><strong>Step 2:</strong> Rationalize by multiplying by the conjugate:<br>√(u + 8) - √(u - 7) = 1<br>Multiply by [√(u + 8) + √(u - 7)]:<br>(u + 8) - (u - 7) = √(u + 8) + √(u - 7)<br>15 = √(u + 8) + √(u - 7)</p><p><strong>Step 3:</strong> From the original equation: √(u + 8) - √(u - 7) = 1. Adding this to 15 = √(u + 8) + √(u - 7):<br>2√(u + 8) = 16 → √(u + 8) = 8 → u + 8 = 64 → u = 56</p><p><strong>Step 4:</strong> Substitute back: 5x² - 6x = 56<br>5x² - 6x - 56 = 0<br>Using the quadratic formula or factoring: (5x + 14)(x - 4) = 0<br>x = 4 or x = -14/5</p><p><strong>Step 5:</strong> Verify both in the original equation:<br>For x = 4: √(80 - 24 + 8) - √(80 - 24 - 7) = √64 - √49 = 8 - 7 = 1 ✓<br>For x = -14/5: √(56 + 84/5 + 8) - √(56 + 84/5 - 7) = √64 - √49 = 8 - 7 = 1 ✓</p><p>∴ Answer: x = 4, -14/5</p>
Correct Answer: x = 4, -14/5