Quadratic Equations
Nature of roots
Grade 11
Question:
<p>The discriminant of a quadratic equation is <\(4 - 12 < 0\) and \(1, 2, 3 \in \mathbb{R}\). If the equation has complex conjugate roots, and all coefficients must be proportional \(\frac{a}{1} = \frac{b}{2} = \frac{c}{3}\), then \(a : b : c\) is:</p>
<p>\(1 : 2 : 3\)</p>
<p>\(1 : 3 : 2\)</p>
<p>\(2 : 3 : 1\)</p>
<p>\(3 : 2 : 1\)</p>
Step-by-Step Solution
Key Concept: The discriminant Δ = b² - 4ac determines the nature and number of roots of a quadratic equation ax² + bx + c = 0 without solving it explicitly.
<p><strong>Key Information:</strong> For a quadratic equation ax² + bx + c = 0, the discriminant is defined as the expression under the square root in the quadratic formula.</p><p><strong>Step 1:</strong> From the quadratic formula: x = (-b ± √(b² - 4ac))/(2a)</p><p><strong>Step 2:</strong> The discriminant Δ = b² - 4ac determines:</p><ul><li>If Δ > 0: Two distinct real roots</li><li>If Δ = 0: One repeated real root (equal roots)</li><li>If Δ < 0: Two complex conjugate roots (no real roots)</li></ul><p><strong>Step 3:</strong> The discriminant is always expressed as <strong>Δ = b² - 4ac</strong> regardless of the specific equation.</p><p>∴ Answer: A</p>
Correct Answer: A