Quadratic Equations
Common roots
Grade 11
Question:
<p>Since \(2x^2 + 3x + 4 > 0\) (as discriminant \(= 9 - 32 = -23 < 0\)) and the coefficients 2, 3, 4 are real, both roots are imaginary conjugate pairs and are common. If \(a : b : c\) is the ratio of coefficients, then \(a : b : c\) equals:</p>
<p>\(2 : 3 : 4\)</p>
<p>\(1 : 2 : 3\)</p>
<p>\(3 : 2 : 1\)</p>
<p>\(4 : 3 : 2\)</p>
Step-by-Step Solution
Key Concept: When a quadratic has negative discriminant and positive leading coefficient, it's always positive. This means inequalities involving such quadratics can be solved by analyzing only the other factors or expressions in the problem.
<p><strong>Step 1:</strong> Analyze the quadratic $2x^2 + 3x + 4$.</p><p>Discriminant: $\Delta = 3^2 - 4(2)(4) = 9 - 32 = -23 < 0$</p><p><strong>Step 2:</strong> Since $\Delta < 0$ and leading coefficient $a = 2 > 0$, the quadratic is always positive for all real $x$.</p><p><strong>Step 3:</strong> Therefore, the inequality $2x^2 + 3x + 4 > 0$ is true for all $x \in \mathbb{R}$.</p><p><strong>Step 4:</strong> If this quadratic appears in a compound inequality or fraction, it does not restrict the solution set and can be effectively 'removed' from consideration when finding the final answer.</p><p>∴ Answer: A</p>
Correct Answer: A