$\Delta = b^2 - 4ac$; if $\Delta > 0$, two distinct real roots; if $\Delta = 0$, one repeated real root; if $\Delta < 0$, two complex conjugate roots

$\Delta = b^2 + 4ac$; determines the number of real roots

$\Delta = c^2 - b^2$; used to find roots

Discriminant is only defined for monic polynomials