<p>Two equations have irrational roots occurring in pairs such that \(\dfrac{51}{3} = 17 = \dfrac{m}{b} = \dfrac{c}{a}\). Find \(\dfrac{c}{a}\).</p>
Step-by-Step Solution
Key Concept: When irrational roots occur in conjugate pairs for quadratic equations with rational coefficients, the sum and product of roots remain rational. Use Vieta's formulas with the given ratio conditions to establish relationships between coefficients.
<p><strong>Step 1:</strong> Recognize that for two quadratic equations with irrational roots occurring in conjugate pairs, if the roots are α±β√k and γ±δ√k respectively, Vieta's formulas ensure the sum and product of roots are rational.</p><p><strong>Step 2:</strong> Observe the given equality: 51/3 = 17 = m/b = c/a. This directly states that the desired ratio c/a equals 17.</p><p><strong>Step 3:</strong> The fraction 51/3 simplifies to 17, confirming the consistency of the given ratios for both equations. Since irrational roots occur in pairs (conjugate pairs), the coefficients maintain proportional relationships.</p><p><strong>Step 4:</strong> From the chain equality, c/a = 17.</p><p>∴ Answer: <strong>17</strong> or <strong>B</strong></p>
Correct Answer: B