<p>For the equation <span class="math">\(4x^2 - 16x + c = 0\)</span>, find the range of <span class="math">\(c\)</span> such that both roots are real and lie in the interval <span class="math">\((1, 3)\)</span>.</p>
Step-by-Step Solution
Key Concept: Use sign conditions on the quadratic function at boundary points and the discriminant to ensure roots lie within an interval.
<p><strong>Step 1:</strong> Rewrite as <span class="math">$x^2 - 4x + \frac{c}{4} = 0$</span></p><p><strong>Step 2:</strong> Let <span class="math">$f(x) = x^2 - 4x + \frac{c}{4}$</span></p><p><strong>Step 3:</strong> For both roots in <span class="math">$(1,3)$</span>, we need:</p><p><strong>Case I (Discriminant):</strong> <span class="math">$D > 0 \Rightarrow 16 - c > 0 \Rightarrow c < 16$</span></p><p><strong>Case II (Vertex condition):</strong> <span class="math">$f(1) > 0 \Rightarrow 1 - 4 + \frac{c}{4} > 0 \Rightarrow \frac{c}{4} > 3 \Rightarrow c > 12$</span></p><p><strong>Case III:</strong> <span class="math">$f(3) < 0 \Rightarrow 9 - 12 + \frac{c}{4} < 0$</span> (for roots to straddle 3 or lie before it)</p><p><strong>Step 4:</strong> Combining all conditions: <span class="math">$12 < c < 16$</span></p><p>∴ Answer is <strong>12 < c < 16</strong>.</p>
Correct Answer: 12 < c < 16