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

$\Rightarrow$ $\alpha$ and $\beta$ are zeros of polynomial $a{x}^2+bx+c=0$

$\Rightarrow$ We know, $\alpha .\beta =\frac{c}{a}$ ---- ( 1 )

$\Rightarrow$ $\alpha +\beta =\frac{-b}{a}$ ----- ( 2 )

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

$\Rightarrow$ ${\left(\frac{-b}{a}\right)}^2={\alpha }^2+{\beta }^2+2\times \frac{c}{a}$ [ Using ( 1 ) and ( 2 ) ]

$\Rightarrow$ $\frac{b^2}{a^2}={\alpha }^2+{\beta }^2+\frac{2c}{a}$

$\Rightarrow$ ${\alpha }^2+{\beta }^2=\frac{b^2}{a^2}-\frac{2c}{a}$

$\Rightarrow$ ${\alpha }^2+{\beta }^2=\frac{b^2-2ac}{a^2}$