Discriminant as intersection counter
SAT glossary · Course Foundations
Definition
On the equation formed by setting two curves equal, Δ answers "how many solutions" by itself — Δ>0 two, Δ=0 one (tangency), Δ<0 none. If the stem asks only for a count, computing Δ is the whole item and solving is wasted time; if it asks for points, report ordered pairs, not x-values.
Discriminant as intersection counter in plain English
When you set two equations equal to each other to find where their graphs cross, the discriminant (Δ = b^2-4ac) of the resulting equation instantly tells you HOW MANY times they intersect, with zero extra work: a positive discriminant means two crossing points, exactly zero means they just barely touch at one point (tangent), and negative means they never meet at all. So if a question only asks 'how many solutions/intersections are there,' you can stop the moment you calculate the discriminant — you don't need to actually find where they cross. But if the question asks for the actual intersection POINTS, you do need to keep going and report full coordinate pairs (x,y), not just the x-values.
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