Strict and inclusive sign conditions

For a 0, ax2+bx+c>0 for every real x exactly when a>0 and D<0. Replacing >0 by 0 changes the condition to a>0, D 0. Negative versions require a<0. The leading coefficient chooses the direction of the parabola; the discriminant decides whether it crosses, touches or misses the axis. Strict positivity cannot include a touching point. Check the leading coefficient and any vanishing case, calculate the discriminant, impose the correct strict or inclusive bound, and combine the parameter conditions. Example x2+4x+c=(x+2)2+c-4. It is positive for every real x when c>4, and never negative when c 4. At c=4 it reaches zero. The diagram shows c=5, with minimum point (-2,1). Exam check: D<0 alone does not mean positive: -x2-1 also has no real roots but is always negative.