Quadratic inequalities of the form $ax^{2+bx+c}$ > 0 are solved by finding the roots and analyzing the sign of the parabola. For a>0 (upward parabola): the expression is positive outside roots (x < alpha or x > beta) and negative between roots (alpha < x < beta). For a<0: signs reverse. When discriminant D<0 (no real roots): expression has the same sign as 'a' everywhere — positive definite if a>0, negative definite if a<0. When D=0: one repeated root, expression touches zero but doesn't change sign. Always convert to standard form first, then check the sign of 'a' and discriminant. For strict inequalities (>), exclude root points; for non-strict (>=), include them. Quadratic inequalities appear as sub-problems in domain finding, range determination, and optimization.