Δ>0 gives two different roots, Δ=0 means the two roots merge into one, and Δ<0 means there's no root among the real numbers.
Module goal. Push the completing-the-square process through once and for all into a universal formula, and use the expression under the root to judge the roots in advance.
eastmath.com · 11.3 The Quadratic Formula and the Discriminant · 11.3.4 Using Δ to Judge the Roots