Is there a relation between the polynomial discriminant and the formula used to solve equations upto degree 4?

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For a quadratic polynomial the polynomial discriminant appears in the quadratic formula however, I found this was not the case for cubic and quartic polynomials. The expressions used in the cubic and quartic formulae are different from the polynomial discriminants or at least they seem different to me. So is it just a coincidence that the polynomial discriminant of a quadratic polynomial appears in the quadratic formula or I have made a mistake somewhere and an alternate form of the discriminant is used in the cubic and quartic formulae. If the discriminants do exist in the cubic and quartic formulae then, is there a relation between them.