<p><strong>Ex. 83:</strong> Sum of all values of <i>x</i> satisfying the equation <i>x</i> = \( \sqrt[]{4 + \sqrt[]{4 + \sqrt[]{4 + \cdots}}} \) is</p>
Step-by-Step Solution
Key Concept: Recognize that an infinite nested radical of the form \( \sqrt{a + \sqrt{a + \sqrt{a + \cdots}}} \) satisfies x = √(a + x), which reduces to a quadratic equation.
<p><strong>Step 1:</strong> Let <i>x</i> = \( \sqrt[]{4 + \sqrt[]{4 + \sqrt[]{4 + \cdots}}} \)</p><p><strong>Step 2:</strong> Since the expression is infinite and self-similar: <i>x</i> = \( \sqrt[]{4 + x} \)</p><p><strong>Step 3:</strong> Squaring both sides: 9<i>x</i>² = 4 + 9<i>x</i></p><p><strong>Step 4:</strong> Rearranging: 9<i>x</i>² - 9<i>x</i> - 4 = 0</p><p><strong>Step 5:</strong> Factoring: (3<i>x</i> - 4)(3<i>x</i> + 1) = 0</p><p><strong>Step 6:</strong> Solutions: <i>x</i> = 4/3 and <i>x</i> = -1/3 (rejected as x must be positive)</p><p><strong>Step 7:</strong> Therefore <i>x</i> = 4/3</p>
Correct Answer: B