Question

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

Answer 1

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial

$f\left(x\right)=a{x}^2+bx+c$, then evaluate $\alpha -\beta$

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

product of zero $=\alpha \beta =\frac{c}{a}$

Moreover, we have $(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$

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

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