Question

if alpha and beta are the zeros of the quadratic polynomial pX square + qX + r then find the value of Alpha square beta + beta square Alpha?

Answer 1

Given:

$p {x}^{2} + qx + r$
here,

a= p, b =q and c=r,

we know that,

sum of zeroes = -b/a

$\alpha + \beta = \frac{ - q}{p}$
_________________________(1)

also,

Product of zeroes= c/a

$\alpha \beta = \frac{r}{p}$
_________________________(2)

now,

$\alpha^{2} \beta + { \beta }^{2} \alpha \\ \\ = \alpha \beta ( \alpha + \beta )$
from (1) and (2)

$\frac{r}{p} ( \frac{ - q}{p} ) \\ \\ = \frac{ - qr}{ {p}^{2} }$