Math PicnicFree printable practice
  • Free
  • Grade 9
  • a = 1, two, one or none

The discriminant worksheets

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.

Customize Practice Save as PDF New questions

Kid stuck on one? You don’t just get the sheet.

See it worked step by step →Answer it on screen →
  • Answer key included
  • Reopen this exact sheet
  • Letter & A4 ready
Ready to printThe discriminant
The discriminant worksheet — a = 1, two, one or none, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

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 this sheet, worked through a step at a time.

x² + 12x + 36 = 0

x² + 12x + 36 = 0
b²−4ac
a = 1b = 12c = 36
12² − 4 × 1 × 36
144 − 144=0
b² − 4ac = 0
How many?One
  1. You do not have to solve this to answer it. One number decides how many real solutions a quadratic has, and it is b² − 4ac — the part that ends up under the square root sign when you do solve it.
  2. Read off a, b and c from x² + 12x + 36 = 0: a = 1, b = 12, c = 36. Write all three in, even the a — on this sheet it is 1, and forgetting it is the mistake that only shows up when it is not 1.
  3. Work it out: 12² is 144, and 4 × 1 × 36 is 144. That leaves 144 − 144 = 0.
  4. 0 is exactly zero, and the square root of zero is zero — adding it and subtracting it land on the same place. So the two answers collapse into ONE.

Pick another skill