<p>The product of all values of <i>x</i> satisfying the equation</p><p>\[\frac{1}{x^2+2x} + \frac{1}{x^2+6x+8} + \frac{1}{x^2+10x+24} = \frac{1}{5} - \frac{1}{x^2+14x+48}\]</p><p>is</p>
Step-by-Step Solution
Key Concept: Recognize telescoping series pattern in partial fractions; most intermediate terms cancel, leaving a simpler equation to solve.
<p><strong>Step 1:</strong> Factor each denominator:</p><p>\(x^2+2x = x(x+2)\)</p><p>\(x^2+6x+8 = (x+2)(x+4)\)</p><p>\(x^2+10x+24 = (x+4)(x+6)\)</p><p>\(x^2+14x+48 = (x+6)(x+8)\)</p><p><strong>Step 2:</strong> Use partial fractions. Each term can be written as a telescoping series:</p><p>\(\frac{1}{(x)(x+2)} = \frac{1}{2}\left(\frac{1}{x} - \frac{1}{x+2}\right)\)</p><p>\(\frac{1}{(x+2)(x+4)} = \frac{1}{2}\left(\frac{1}{x+2} - \frac{1}{x+4}\right)\)</p><p>\(\frac{1}{(x+4)(x+6)} = \frac{1}{2}\left(\frac{1}{x+4} - \frac{1}{x+6}\right)\)</p><p>\(\frac{1}{(x+6)(x+8)} = \frac{1}{2}\left(\frac{1}{x+6} - \frac{1}{x+8}\right)\)</p><p><strong>Step 3:</strong> Left side becomes:</p><p>\(\frac{1}{2}\left(\frac{1}{x} - \frac{1}{x+8}\right) = \frac{1}{5} - \frac{1}{2}\left(\frac{1}{x+6} - \frac{1}{x+8}\right)\)</p><p><strong>Step 4:</strong> Simplify and solve for <i>x</i>. After cancellation and algebra, this leads to a quadratic with roots whose product is –20.</p><p>∴ Answer is D.</p>
Correct Answer: D