Question

If α and β are the zeros of the quadratic polynomial f(x) = x² – x – 4, find the value of \(\frac{1}{α} + \frac{1}{β} – αβ\).

Answer 1

From the question, it’s given that: 

α and β are the roots of the quadratic polynomial f(x) where a = 1, b = -1 and c = – 4 

Sum of the roots = α + β = \(\frac{-b}{a}\) 

= – \(\frac{(-1)}{1}\) 

= 1 

Product of the roots = αβ = \(\frac{c}{a}\) 

= \(\frac{-4}{1}\) 

= – 4 

\(\frac{1}{α} + \frac{1}{β} – αβ\)

⇒ \([\frac{(α +β)}{ αβ}] – αβ\) 

⇒ [\(\frac{(1)}{(-4)}\)] – (-4) 

= \(\frac{-1}{4}\) + 4 

= \(\frac{15}{4}\)