<p>If \(a, b, c\) are three distinct positive real numbers, then the number of real roots of \(ax^2 + 2b|x| - c = 0\) is</p>
Step-by-Step Solution
Key Concept: Since the equation contains |x|, substituting y = |x| ≥ 0 transforms it into ay² + 2by - c = 0. Each positive root in y yields two roots in x (±y), while y = 0 gives only x = 0.
<p><strong>Step 1:</strong> Substitute y = |x| where y ≥ 0. The equation becomes: ay² + 2by - c = 0</p><p><strong>Step 2:</strong> Apply quadratic formula: y = (-2b ± √(4b² + 4ac))/(2a) = (-b ± √(b² + ac))/a</p><p><strong>Step 3:</strong> Since a, b, c > 0, we have √(b² + ac) > b, so:</p><ul><li>y₁ = (-b + √(b² + ac))/a > 0 ✓ (valid)</li><li>y₂ = (-b - √(b² + ac))/a < 0 ✗ (rejected, as y ≥ 0)</li></ul><p><strong>Step 4:</strong> The single positive root y₁ in the y-equation gives exactly TWO real roots in x: x = ±y₁</p><p>∴ Answer: <strong>C</strong> (2 real roots)</p>
Correct Answer: C