When the sign does not change

A repeated root is a touching point. In a(x-h)2, the square is positive for x!= h and zero at x=h. A quadratic with no real root keeps one sign on the entire real line. A square never becomes negative, so a graph that only touches an axis does not cross it. Adding a positive constant to an upward square leaves it above the axis. Complete the square or classify the roots; determine whether zero is attained; then test the requested strict or inclusive sign. State “all real numbers”, “no solution”, a single value or excluded value explicitly. Example (x-1)2 0 gives only x=1; (x-1)2>0 gives x 1. The expression (x-1)2+2 is positive for every real x. Exam check: the two-distinct-roots interval rule cannot be used when roots coincide or are not real.