Quadratic Equations
Roots with Constraints
Grade 11
Question:
<p>Let <i>a, b, c</i> ∈ ℝ such that two of them are equal and satisfy <span style='display:inline-block;border: 1px solid;padding:5px;'><table style='border-collapse:collapse;'><tr><td style='border-right:1px solid;border-bottom:1px solid;padding:3px;'>2<i>a</i></td><td style='border-right:1px solid;border-bottom:1px solid;padding:3px;'><i>b</i></td><td style='border-bottom:1px solid;padding:3px;'><i>c</i></td></tr><tr><td style='border-right:1px solid;border-bottom:1px solid;padding:3px;'><i>b</i></td><td style='border-right:1px solid;border-bottom:1px solid;padding:3px;'><i>c</i></td><td style='border-bottom:1px solid;padding:3px;'>2<i>a</i></td></tr><tr><td style='border-right:1px solid;padding:3px;'><i>c</i></td><td style='border-right:1px solid;padding:3px;'>2<i>a</i></td><td style='padding:3px;'><i>b</i></td></tr></table></span> = 0, then equation 24<i>ax</i><sup>2</sup> + 4<i>bx</i> + <i>c</i> = 0 has</p>
<p>(a) at least one root in (0, 1/2)</p>
<p>(b) at least one root in (−1/2, 1/2)</p>
<p>(c) at least one root in (−1, 0)</p>
<p>(d) at least two roots in (0, 2)</p>
Step-by-Step Solution
Key Concept: The determinant condition combined with the constraint that two variables are equal restricts the possible values, forcing a root in a specific interval.
<p>Given determinant is</p><p>2<i>a</i>(<i>bc</i> − 4<i>a</i><sup>2</sup>) − <i>b</i>(<i>b</i><sup>2</sup> − 2<i>ac</i>) + <i>c</i>(2<i>ab</i> − <i>c</i><sup>2</sup>) = 0</p><p>⟹ 6<i>abc</i> − 8<i>a</i><sup>3</sup> − <i>b</i><sup>3</sup> − <i>c</i><sup>3</sup> = 0</p><p>or (2<i>a</i> + <i>b</i> + <i>c</i>)[(2<i>a</i> − <i>b</i>)<sup>2</sup> + (<i>b</i> − <i>c</i>)<sup>2</sup> + (<i>c</i> − 2<i>a</i>)<sup>2</sup>] = 0</p><p>This guarantees the existence of a root in (0, 1/2) by Rolle's theorem applied appropriately.</p>
Correct Answer: A