Assess students' ability to compute and interpret the discriminant (b^2 - 4ac) of quadratic equations to determine root types (two distinct real roots, one repeated real root with multiplicity 2, or two complex conjugate roots), explain implications for factorability, and connect discriminant signs to graph behavior (number of x-intercepts, vertex location relative to the x-axis, and effect of a's sign). Include problems requiring numeric computation, sign reasoning, and short explanations or sketches.