Quadratic Equations
Nature of roots
Grade 11
Question:
<p>If \(f(x) = ax^2 + bx + c\), where \(a \neq 0\), \(b, c \in \mathbb{R}\), then which of the following conditions implies that \(f(x)\) has real roots?</p>
<p>\(a + b + c = 0\)</p>
<p>\(a\) and \(c\) are of opposite signs</p>
<p>\(4ac - b^2 < 0\)</p>
<p>\(a\) and \(b\) are of opposite signs</p>
Step-by-Step Solution
Key Concept: A quadratic has real roots if and only if its discriminant Δ = b² - 4ac ≥ 0. You must identify ALL conditions that guarantee this inequality holds, not just one.
<p><strong>Key Principle:</strong> f(x) has real roots ⟺ Δ = b² - 4ac ≥ 0</p><p><strong>Step 1:</strong> Analyze each option systematically using the discriminant condition.</p><p><strong>Step 2:</strong> For option A: If b² ≥ 4ac (directly given), then Δ ≥ 0 ✓</p><p><strong>Step 3:</strong> For option B: If c and a have opposite signs (c·a < 0), then -4ac > 0, so b² - 4ac ≥ 0 regardless of b ✓</p><p><strong>Step 4:</strong> For option C: If |b| ≥ 2|a|·|c|^(1/2), then b² ≥ 4|a||c| ≥ 4ac (when a,c same sign) or > 4ac (when opposite signs) ✓</p><p><strong>Step 5:</strong> Check options claiming f(x) > 0 for all x or f(x) ≤ k: These require Δ < 0 (no real roots), so they contradict our requirement.</p><p>∴ Answer: A,B,C</p>
Correct Answer: A,B,C