Question

if alpha and beta are the zeroes of quadratic polynomial f(x)=x^2-x-4 find the value of 1/alpha+1/beta-alphabeta

Answer 1

Given:

$f(x) = {x}^{2} - x - 4$
here,

a= 1, b= -1 and c= -4,

we know that,

sum of zeroes= -b/a

$\alpha + \beta = 1$
____________________(1)

also,

Product of zeroes= c/a

$\alpha \beta = - 4$
_______________________(2)

now,

$\frac{1}{ \alpha } + \frac{1}{ \beta } - \alpha \beta \\ \\ = \frac{ \beta + \alpha }{ \alpha \beta } - \alpha \beta \\ \\ = \frac{ \beta + \alpha - \alpha \beta ( \alpha \beta )}{ \alpha \beta }$
substituting the values of (1) and (2),

$\frac{1 + 4( - 4)}{ - 4 } \\ \\ = \frac{1 - 16}{ - 4} \\ \\ = \frac{ - 15}{ - 4} \\ \\ = \frac{15}{4}$