double root

A double root is a root that, in a sense, counts twice — the two solutions of a quadratic have merged into a single repeated value. Instead of crossing the x-axis at two separate points, the parabola just kisses the axis at one point and turns back. That touch point is the double root.

This happens exactly when the discriminant is zero: b^2 - 4ac = 0. In the quadratic formula the ± part vanishes (you add and subtract zero), so both branches give the same answer, x = -b/(2a). In factored form the quadratic is a perfect square, a(x - r)^2, with r appearing twice.

Calling it a “double” root is more than bookkeeping: it makes counting roots consistent. The fundamental theorem of algebra says a degree-n polynomial has exactly n roots counted with multiplicity, and a double root contributes two to that count. So a quadratic always has “two” roots — they just sometimes coincide.