Question

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$, then evaluate
${\alpha }^2+{\beta }^2$

Answer 1

Consider the given quadratic polynomial.

$f\left(x\right)=a{x}^2+bx+c$ …….. (1)

Since, $\alpha ,\beta$ are the zeroes of given polynomial.

So,

$\alpha +\beta =-\frac{b}{a}$

$\alpha \beta =\frac{c}{a}$

Here,

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

$\Rightarrow {(\alpha +\beta )}^2-2\alpha \beta$

$\Rightarrow {(-\frac{b}{a})}^2-\frac{2c}{a}$

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

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

Hence, this is the answer.