Find the sum of roots of the polynomial

Question

If $\alpha$ and $\beta$ are the zeros of a quadratic polynomial, what is the value of $\alpha - \beta$ ?

I have to find the answer in terms of the coefficients of the polynomial and I have to use the formulas $\alpha + \beta = -b/a,\alpha\cdot\beta=c/a$.

Answer 1

If $\alpha,\beta$ are the roots of the polynomial $$ p(x)=x^2-sx+p = (x-\alpha)(x-\beta)\tag{1}$$ we have $s=\alpha+\beta, p=\alpha\beta$ (Viète's formulas). It follows that: $$ (\alpha-\beta)^2 = (\alpha+\beta)^2-4\alpha\beta = s^2-4p\tag{2} $$ so the distance between the roots is the square root of the discriminant.