If α and β are the zeros of the quadratic polynomial f(x) = x^2 – 5x + 4, find the value of 1/α + 1/β – 2αβ.

Question

If α and β are the zeros of the quadratic polynomial f(x) = x² – 5x + 4, find the value of \(\frac{1}{α} + \frac{1}{β} – 2αβ.\)

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 = -5 and c = 4

Sum of the roots = α + β = \(\frac{-b}{a}\) = – \(\frac{(-5)}{1}\) = -5 

Product of the roots = αβ = \(\frac{c}{a}\) = \(\frac{4}{1}\) = 4 

\(\frac{1}{α} + \frac{1}{β} – 2αβ.\)

⇒ \([\frac{(α +β)}{αβ}]\) – 2αβ 

⇒ \(\frac{(-5)}{4}\) – 2(4) 

= \(\frac{-5}{4}\) – 8 

= \(\frac{-27}{4}\)