When the leading coefficient is not 1, b² − 4ac still decides it — but the a can no longer be ignored. These free printable worksheets use the discriminant with a real coefficient in front.
Kid stuck on one? You don’t just get the sheet.
The answer key prints on a separate page.
With a = 1 a child can compute b² − 4c, forget the a entirely, and still be right. That habit breaks here. For 3x² + 4x + 2 the discriminant is 4² − 4·3·2 — the a is a factor of the 4ac term, and dropping it changes both the number and, often, the answer. Writing all three letters in is the whole discipline of this page.
The three cases read the same as always: b² − 4ac above zero means two real solutions, zero means one, below zero means none, because the last step of solving is a square root. What changes is only the middle of the calculation — 4ac is now a product of three numbers, not two — so this is the discriminant with the one shortcut removed.
One question from this sheet, worked through a step at a time.
2x² − 16x − 7 = 0