Quadratic Equations
Discriminant Analysis
Grade 11
Question:
<p>If <strong>f</strong>(<strong>x</strong>) = <strong>ax</strong><sup>2</sup> − <strong>bx</strong> + <strong>c</strong> has two distinct roots α and β, and f(0) and f(1) are of the same sign, with the constraint α(1 − α) ≤ 1/4, find the least value of <strong>b</strong>.</p>
Step-by-Step Solution
Key Concept: The discriminant condition b² ≥ 4ac combined with minimum values of a and c determines the minimum value of b.
<p><strong>Step 1:</strong> From the analysis of a, we established that a ≥ 5</p><p><strong>Step 2:</strong> From the discriminant condition b² − 4ac ≥ 0, we need b² ≥ 4ac</p><p><strong>Step 3:</strong> With a ≥ 5 and c ≥ 1 (from earlier steps), we get b² ≥ 4(5)(1) = 20</p><p><strong>Step 4:</strong> Therefore b ≥ √20 ≈ 4.47. Since b ∈ ℤ, we have b ≥ 5</p><p>∴ <strong>Least value of b is 5</strong></p>
Correct Answer: 5