What does α - β equal if a quadratic equation has roots of α and β

Question

Considering the following equation $$px^2 - qx - r = 0$$ whose roots are $\alpha$ and $β$. What would $α - β$ be in terms of $p$, $q$, and $r$?

I understand that I have to use the sum and product of roots, and have found both in terms of $p$, $q$, and $r$:

$$α + β = \frac{q}{p}$$ $$αβ = -\frac{r}{p}$$

I do not know how I would use this to find $α - β$ though, any help is appreciated, thanks!

Answer 1

Hint: $$(\alpha-\beta)^2 = (\alpha+\beta)^2-4\alpha \beta.$$

Answer 2

your assumption $\alpha - \beta = \frac{q}{p}$ is wrong. $$ \alpha + \beta = \frac{q}{p} $$ is correct one then you can use the formula $(\alpha - \beta)^2=(\alpha + \beta)^2-4\alpha \beta$ and you can solve for $\alpha - \beta$.

Answer 3

$$(\alpha-\beta)^2 = (\alpha+\beta)^2-4\alpha \beta = \dfrac{4pr +q^2}{p^2} $$ $$ \alpha -\beta =\pm \dfrac{\sqrt{ 4pr +q^2}}{{p}}$$ Actually you can write out the qudratic roots separately and subtract one from the other.. even if it appears brute force. The discriminant is an important part of the result. (It vanishes for equal roots).