Question

If $\alpha and\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=x^2-5x+4,$ then the value of $\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta$ is

• A.
  $\frac{-27}{4}$
  
• B.
  $\frac{27}{4}$
  
• C.
  $\frac{25}{2}$
  
• D.
  $\frac{-25}{2}$
  

Answer 1

The correct option is A

$\frac{-27}{4}$

$f\left(x\right)=x^2-5x+4\alpha +\beta =-\frac{b}{a}=-\frac{(-5)}{1}=5\alpha \beta =\frac{c}{a}=\frac{4}{1}=4\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta =\frac{\beta +\alpha -2(\alpha \beta {)}^2}{\alpha \beta }=\frac{5-2(4{)}^2}{4}=\frac{5-32}{4}=-\frac{27}{4}\therefore \frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta =\frac{-27}{4}$
Hence, the correct answer is option (a)