Quadratic Equations Worksheets

Every answer is a pair of integers

The equations are built backwards. Two whole-number roots are chosen first, then multiplied out into a trinomial — so every problem factorises cleanly and every answer on the key is exact.

No surds, no decimals, nothing that needs rounding. That is a teaching decision: a pupil learning the method should not be simultaneously fighting an irrational answer and wondering whether they have made an arithmetic slip.

The two modes

Factoring gives you x² + bx + c = 0, with roots drawn from −9 to 9. The leading coefficient is 1, which is the case the factorising method is taught on.

Formula gives you ax² + bx + c = 0, where a is 2, 3 or 4 and the roots come from −6 to 6. A leading coefficient breaks the simple "two numbers that multiply to c and add to b" shortcut, so this is the set that needs the quadratic formula or completing the square — and the roots are still integers, so the discriminant always comes out a perfect square.

Teaching order

Factoring first, and for longer than feels necessary.

The formula works on everything and is therefore tempting to teach immediately, but a pupil who reaches for it on x² + 5x + 6 has stopped seeing structure and started substituting into a machine. Factorising builds the number sense — which pairs multiply to six, which of them add to five — that makes the rest of algebra readable.

Move to the formula mode once factorising is fluent, or when a is not 1.

Checking without the key

Both roots substitute back into the original equation and give zero. Teach that as the habit and half the marking disappears.

There is a second check worth knowing: for x² + bx + c, the roots sum to −b and multiply to c. A pupil who has found 2 and 3 for x² − 5x + 6 can confirm both facts in about four seconds, which is quicker than substituting and catches sign errors, the commonest mistake by a distance.

Signs

Because the roots range across zero, negative roots turn up constantly and so do the sign patterns that trip people up. x² + x − 6 factorises to (x + 3)(x − 2), and the number of pupils who write (x − 3)(x + 2) is not small.

That is the point of drilling it, and the answer key catches it every time.

Practicalities

Twelve problems is about right for a lesson; four makes a starter and thirty makes a revision sheet. Print the key or leave it off if the sheet is going home as homework.