Question

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0,(a\ne 0)$ then find the values of ${\alpha }^2\beta +\alpha {\beta }^2$

Answer 1

We have,

value of ${\alpha }^2+{\beta }^2$

The polynomial is $a{x}^2+bx+c$ whose zeroes are $\alpha$ and $\beta$

Thus sum of the zeroes $=\alpha +\beta =\frac{-b}{a}$

The product of the zeroes $=\alpha \times \beta =\frac{c}{a}$

$\therefore {\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2=2\alpha \beta$

$={\left(\frac{-b}{a}\right)}^2=2\times \frac{c}{a}$

$=\frac{b^2}{a^2}-\frac{2c}{a}$

$=\frac{b^2-2ac}{a^2}$

Hence, this is answer.