Quadratic Equations Worksheets

What this quadratic equations generator does

Every worksheet is built to have clean integer roots. Instead of picking random coefficients and hoping the equation factors, the generator starts with two whole-number roots and multiplies them out. In factoring mode that means monic trinomials of the form x² + bx + c = 0 where (x − r₁)(x − r₂) reproduces the equation exactly. In formula mode it multiplies the factored form by a leading coefficient of 2, 3, or 4 to produce ax² + bx + c = 0, so the quadratic formula is genuinely needed but the discriminant is still a perfect square and the answers stay whole.

Choosing between factoring and formula mode

Factoring mode is the right starting point once students know their times tables and can spot factor pairs. All equations are monic (leading coefficient 1), so the task is to find two numbers that multiply to c and add to b. Formula mode introduces a leading coefficient greater than one, which makes factoring by inspection harder and gives students a natural reason to reach for x = (−b ± √(b² − 4ac)) / 2a. Assigning both modes across a unit lets learners see how the two methods connect.

What you can customise

• Mode: factoring (x² + bx + c = 0) or formula (ax² + bx + c = 0)
• Problem count: 4 to 30 equations per sheet, laid out in two columns
• Answer key: toggle a second page listing both roots for every equation
• Name and date fields: add a header line for classroom hand-in, or hide it for a clean look
• Title: set your own heading or keep the default

How to use the tool

1. Pick factoring or formula mode.
2. Set how many equations you want on the sheet.
3. Toggle the answer key on if you want a marking page.
4. Decide whether to show Name and Date fields.
5. Preview updates live; click Download to save the branded PDF.
6. Print on A4 or US Letter and hand out.

Worked example

In factoring mode the generator might choose roots 2 and 3. Multiplying (x − 2)(x − 3) gives x² − 5x + 6 = 0, and the answer key reads x = 2 or x = 3. In formula mode it might pick a = 2 with roots −1 and 4: 2·(x + 1)(x − 4) expands to 2x² − 6x − 8 = 0, and the answer key again gives x = −1 or x = 4. Students who apply the formula compute a discriminant of 100, a perfect square, confirming that carefully constructed practice keeps the arithmetic honest.

How it works under the hood

Each equation is generated from its roots rather than its coefficients. For factoring mode the tool picks two non-zero integers r₁ and r₂ between −9 and 9, then sets b = −(r₁ + r₂) and c = r₁·r₂. For formula mode it also picks a leading coefficient a between 2 and 4 and scales the expansion, so b = −a(r₁ + r₂) and c = a·r₁·r₂. This guarantees the discriminant b² − 4ac is a perfect square and both solutions are integers. A seeded random number generator means the same settings can reproduce the same sheet when needed.

Designed for A4 and US Letter printing

The layout fits both A4 (210 × 297 mm) and US Letter (8.5 × 11 in) without clipping. Equations appear in two columns with a solution line beneath each, and margins, header, and footer adjust automatically so any home or office printer produces a clean, readable page with no manual scaling.

Notes and limitations

• All roots are integers by design; the tool does not generate irrational or complex solutions.
• Formula-mode leading coefficients are limited to 2, 3, and 4 to keep coefficients manageable.
• Equations are randomised, so an occasional double root (for example x² − 4x + 4 = 0) can appear; the answer key flags it.
• Problem count caps at 30 to keep two-column equations legible; generate a second sheet if you need more.