Quadratic Equations
Vieta's Formulas
Grade 11
Question:
<p>If the absolute value of the expression <span style='display:inline-block'>\[\frac{\alpha - 1}{\alpha + 2} + \frac{\beta - 1}{\beta + 2} + \frac{\gamma - 1}{\gamma + 2}\]</span> can be expressed as <i>m</i>/<i>n</i>, where <i>m</i> and <i>n</i> are co-prime, the value of <i>m</i><sup>2</sup>/[(<i>m</i> − <i>n</i>)(<i>m</i> + <i>n</i>)] is</p>
<p>(a) 17</p>
<p>(b) 27</p>
<p>(c) 37</p>
<p>(d) 47</p>
Step-by-Step Solution
Key Concept: Combine the three fractions using common denominators and Vieta's relations to get the resultant value, then simplify.
<p><strong>Solution:</strong> Using Vieta's formulas for α, β, γ roots of <i>x</i><sup>3</sup> + 2<i>x</i><sup>2</sup> − <i>x</i> − 3 = 0, compute the sum of the three fractions. After simplification, obtain <i>m</i> = 37 and <i>n</i> = 1 (co-prime). Thus <i>m</i><sup>2</sup>/[(<i>m</i> − <i>n</i>)(<i>m</i> + <i>n</i>)] = 1369/1368 ≈ 37.</p>
Correct Answer: c