discriminant
The discriminant is a single number you can compute from a quadratic's coefficients that tells you what kind of roots to expect — before you bother solving. Its name comes from “to discriminate”, in the sense of distinguishing between cases. It is the part of the quadratic formula hiding under the square root.
For ax^2 + bx + c = 0, the discriminant is D = b^2 - 4ac. Three cases: if D > 0 there are two distinct real roots; if D = 0 there is exactly one real root (a repeated, or double, root); if D < 0 there are no real roots, only a pair of complex-conjugate roots.
Why does this work? In the formula, you take ± sqrt(D). A positive D gives two different square-root values, a zero D collapses them into one, and a negative D forces the square root of a negative number — which is not real. So D alone decides the count and type of roots, which is invaluable when you only need to know how many solutions exist.
For x^2 - 4x + 4 = 0, D = (-4)^2 - 4(1)(4) = 16 - 16 = 0, so there is one double root, x = 2.
D = 0 is the borderline case between two real roots and none.
When D is a perfect square (and a, b, c are integers), the roots are rational and the quadratic factors over the integers — a quick check for whether factoring will work.