Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $p\left(x\right)=4x^2-5x-1$, find the value of ${\alpha }^2\beta +\alpha {\beta }^2$.

Answer 1

We have, $p\left(x\right)=4x^2-5x-1$

On comparing the given equation with the standard quadratic equation $a{x}^2+bx+c=0$, we have

$a=4$, $b=-5$, and $c=-1$

Roots are $\alpha$ and $\beta$

Sum of roots $=\alpha +\beta =\frac{-b}{a}=\frac{5}{4}$

Products of Roots $=\alpha \beta =\frac{c}{a}=\frac{-1}{4}$

We want to find,

${\alpha }^2\beta +{\beta }^2\alpha =\alpha \beta (\alpha +\beta )\left[\text{Taking}\alpha \beta \text{as common}\right]$

$=\left(\frac{-1}{4}\right)\left(\frac{5}{4}\right)=\left(\frac{-5}{16}\right)[\text{on putting the values of}\alpha +\beta ,\alpha \beta ]$

$\therefore {\alpha }^2\beta +{\beta }^2\alpha =\left(\frac{-5}{16}\right)$