One number decides how many real solutions a quadratic has, and only its sign matters: b² − 4ac. These free printable worksheets ask it of a quadratic where a is 1.
You do not have to solve the quadratic — only work out b² − 4ac. With a equal to 1 that is b² − 4c, and its sign is the answer: above zero there are two real solutions, exactly zero gives one, below zero gives none. A child who computes one number can answer a question they could not solve.
Why this sheet works
The reason underneath is worth having instead of the table. Solving a quadratic ends by taking the square root of b² − 4ac: a positive number has two roots, so two answers; zero has one, so the answers land on top of each other; a negative has none, so there is nothing to take the root of. Rebuild that and you never have to memorize which case is which.
How this one works
One question from a sheet like this one, worked a step at a time — with a sentence saying what just happened and why.
