Intuitive understand behind formulae derivations

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For example, I completely understand how to derive the quadratic formula, but I have no idea why one gets what one gets. I obviously understand that -b/2a is the axis of symmetry and that sqrt(b^2-4ac) is the distance from the axis of symmetry to the zeros.
But why? Why would -b/2a give the axis of symmetry for example? Could it be purely derived intuitively?