Quadratic Equations
Quadratic Inequalities
Grade 11
Question:
<p>Complete set of real values of <i>k</i> for which the inequality <i>kx</i>² – <i>kx</i> – 1 < 0 holds for any real <i>x</i>, satisfy</p>
<p>(A) <i>k</i> ∈ (–4, 0)</p>
<p>(B) <i>k</i> ∈ (–4, 0]</p>
<p>(C) <i>k</i> ∈ [–4, 0)</p>
<p>(D) <i>k</i> ∈ [–4, 0]</p>
Step-by-Step Solution
Key Concept: For a quadratic inequality to hold for all real x, either the coefficient is zero (making it a constant inequality) or the quadratic has negative leading coefficient and negative discriminant.
<p><strong>Case 1:</strong> If <i>k</i> = 0, the inequality becomes –1 < 0, which is always true.</p><p><strong>Case 2:</strong> If <i>k</i> ≠ 0, for <i>kx</i>² – <i>kx</i> – 1 < 0 to hold for all <i>x</i>, we need <i>k</i> < 0 and discriminant < 0.</p><p>Discriminant: Δ = <i>k</i>² + 4<i>k</i> < 0 ⟹ <i>k</i>(<i>k</i> + 4) < 0 ⟹ –4 < <i>k</i> < 0.</p><p>Combining both cases: <i>k</i> ∈ [–4, 0).</p><p>∴ Answer is C.</p>
Correct Answer: C