That piece under the root, b²−4ac, is called the discriminant Δ, and it's a weather forecast that tells you ahead of time what the roots will look like.
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 · 11.3.3 The Discriminant Δ=b²−4ac