Quadratic Equations
Sign of Quadratic Expression
Grade 11
Question:
<p>If \(x^2 + 2ax + 10 - 3a > 0\) for all \(x \in \mathbb{R}\), then</p>
<p>(A) \(-5 < a < 2\)</p>
<p>(B) \(a < -5\)</p>
<p>(C) \(a > 5\)</p>
<p>(D) \(2 < a < 5\)</p>
Step-by-Step Solution
Key Concept: A quadratic is always positive if and only if its discriminant is negative.
<p>For \(x^2 + 2ax + 10 - 3a > 0\) for all \(x \in \mathbb{R}\), the discriminant must be negative.</p><p>Discriminant: \(\Delta = (2a)^2 - 4(1)(10 - 3a) < 0\)</p><p>\(4a^2 - 40 + 12a < 0\)</p><p>\(a^2 + 3a - 10 < 0\)</p><p>\((a + 5)(a - 2) < 0\)</p><p>\(-5 < a < 2\)</p>
Correct Answer: A