One number decides it, and only its sign matters: b² − 4ac. These free printable worksheets ask how many real solutions a quadratic has.
If they get stuck
The answer key prints on a separate page.
A student who has only ever solved quadratics that come out meets x² + 2x + 5 = 0, finds no pair of numbers that multiplies to 5 and adds to 2, and assumes they made a mistake. They did not: there is no such pair, and there is no real answer. “None” is a result rather than a failure, and that sentence is the whole reason this sheet exists.
There is one fact underneath the three cases, and it is worth having instead of the table. Solving a quadratic ends by taking the square root of b² − 4ac. A positive number has two square roots, so there are two answers. Zero has one, so the two answers land on top of each other. A negative has none, so there is nothing to take the root of. A student who can rebuild that never has to remember which case is which.
One question from a sheet like this one, worked a step at a time — with a sentence saying what just happened and why.
How many real solutions?, step by step