Quadratic Equations Worksheet Generator
On the easy and medium settings every quadratic factors over the integers, so students can solve by factoring. On hard the generator forces a positive non-square discriminant, so the quadratic formula is required and the roots come out in surd form.
A difficulty setting that changes the method
Quadratics are solved two ways, and a curriculum teaches them in order: factoring first, because it is fast when it works, then the general formula, because it always works. Practice sheets that mix the two indiscriminately teach neither well.
This generator separates them by difficulty. Easy and medium produce quadratics that factor over the integers; hard deliberately produces ones that do not, forcing the formula and leaving the roots in surd form.
The three levels are not the same task with bigger numbers. Easy and medium reward recognising a factorisation, and the answers are whole numbers a student can verify by substitution. Hard removes that route entirely — no integer factorisation exists, so the formula is the only way through.
That makes the setting a curriculum choice rather than a strain dial. A sheet at the wrong level does not just feel harder or easier; it exercises a different skill.
How to use this calculator
- Choose how many problems Between 1 and 30. A larger request is clamped rather than refused, and a non-numeric entry becomes ten.
- Pick the difficulty Easy and medium for factoring practice, hard for the quadratic formula. The choice determines which method the sheet requires.
- Set a seed, or leave it blank A whole number reproduces the sheet exactly. Blank picks one at random and prints it alongside the output.
- Solve, then check the key The problems come first and the key last. On easy and medium the answers can also be verified by substituting them back.
How each level is generated
Easy and medium work backwards from the roots, as the factoring sheets do. Two distinct non-zero integers are drawn — between −6 and 6 on easy, narrowed to between −4 and 4 on medium — and the quadratic is assembled from them, with medium scaling every coefficient by a leading factor between 2 and 4. Because the roots were chosen first, a clean factorisation is guaranteed.
Hard inverts the requirement. It draws a middle coefficient and a constant, both non-zero and between −7 and 7, computes the discriminant, and rejects the pair unless the discriminant is both positive and not a perfect square. A positive discriminant keeps the roots real; a non-square one keeps them irrational, which is exactly what makes factoring useless.
That rejection loop runs up to thirty times. If no suitable pair is found in those attempts — which happens when the draws keep landing on squares or negatives — the generator falls back to constructing a factorable quadratic from two integer roots instead, so a problem is always produced rather than the sheet failing.
The answer key for hard reports the roots in surd form rather than as decimals, with any perfect-square factor pulled out of the radical. That is the form the formula naturally produces and the form an exam expects.
What each input means
- n Number of problems — form field “Number of problems”
- How many equations appear, clamped to between 1 and 30. Units: problems.
- d Difficulty — form field “Difficulty”
- Selects the method the sheet requires: factoring on easy and medium, the quadratic formula on hard.
- s Seed — form field “Seed (optional)”
- Optional whole number fixing the sheet. Reported with the output whether supplied or generated.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
Ten factorable quadratics
Ten easy problems from a fixed seed. Every leading coefficient is one and every root is a whole number.
Inputs Number of problems = 10, Difficulty = easy, Seed (optional) = 20260820
- Seed 20260820 (use this seed to regenerate the same worksheet)
- Count 10 problems
- Problem 1. Solve: x^2 − 5x − 6 = 0
- Problem 2. Solve: x^2 − 6x + 5 = 0
- Problem 3. Solve: x^2 + 2x − 15 = 0
- Problem 4. Solve: x^2 + 6x + 5 = 0
- Problem 5. Solve: x^2 + 7x + 10 = 0
- Problem 6. Solve: x^2 − x − 6 = 0
- Problem 7. Solve: x^2 − 25 = 0
- Problem 8. Solve: x^2 + 2x − 24 = 0
- Problem 9. Solve: x^2 − 5x + 6 = 0
- Problem 10. Solve: x^2 − 2x − 15 = 0
- Answer key 1. x = -1 or x = 6; 2. x = 1 or x = 5; 3. x = -5 or x = 3; 4. x = -5 or x = -1; 5. x = -2 or x = -5; 6. x = 3 or x = -2; 7. x = -5 or x = 5; 8. x = 4 or x = -6; 9. x = 2 or x = 3; 10. x = 5 or x = -3
Result 10-problem worksheet — Quadratic equations (easy) (seed 20260820)
Each equation factors into two linear brackets, so the roots can be read off once the factorisation is found. Substituting an answer back into the equation confirms it without the key.
The two roots are always distinct, since the second is redrawn whenever it matches the first. No problem here has a repeated root or a zero root.
Ten with a leading coefficient
The medium level, where every coefficient carries a common factor between 2 and 4.
Inputs Number of problems = 10, Difficulty = medium, Seed (optional) = 20260820
- Seed 20260820 (use this seed to regenerate the same worksheet)
- Count 10 problems
- Problem 1. Solve: 3x^2 − 15x + 12 = 0
- Problem 2. Solve: 4x^2 + 8x − 32 = 0
- Problem 3. Solve: 2x^2 + 10x + 8 = 0
- Problem 4. Solve: 3x^2 + 18x + 24 = 0
- Problem 5. Solve: 4x^2 + 24x + 32 = 0
- Problem 6. Solve: 4x^2 + 4x − 48 = 0
- Problem 7. Solve: 3x^2 − 18x + 24 = 0
- Problem 8. Solve: 2x^2 − 4x − 16 = 0
- Problem 9. Solve: 4x^2 − 8x − 12 = 0
- Problem 10. Solve: 2x^2 − 2x − 12 = 0
- Answer key 1. x = 4 or x = 1; 2. x = -4 or x = 2; 3. x = -4 or x = -1; 4. x = -2 or x = -4; 5. x = -2 or x = -4; 6. x = 3 or x = -4; 7. x = 2 or x = 4; 8. x = 4 or x = -2; 9. x = 3 or x = -1; 10. x = -2 or x = 3
Result 10-problem worksheet — Quadratic equations (medium) (seed 20260820)
The roots are still whole numbers despite the larger coefficients, because the leading factor multiplies the whole quadratic rather than changing where it crosses zero. Dividing through by it first reduces the problem to the easy case.
Spotting that common factor is the skill this level adds. Missing it means searching among far more candidate factorisations than the problem actually needs.
Ten requiring the formula
The hard level, where the discriminant is positive but never a perfect square.
Inputs Number of problems = 10, Difficulty = hard, Seed (optional) = 20260820
- Seed 20260820 (use this seed to regenerate the same worksheet)
- Count 10 problems
- Problem 1. Solve: x^2 − 6x + 3 = 0
- Problem 2. Solve: x^2 − 6x − 6 = 0
- Problem 3. Solve: x^2 − x − 3 = 0
- Problem 4. Solve: x^2 − 6x + 3 = 0
- Problem 5. Solve: x^2 − 3x − 6 = 0
- Problem 6. Solve: x^2 + 5x + 5 = 0
- Problem 7. Solve: x^2 − 7x + 2 = 0
- Problem 8. Solve: x^2 + 7x − 3 = 0
- Problem 9. Solve: x^2 − 2x − 7 = 0
- Problem 10. Solve: x^2 + 4x − 1 = 0
- Answer key 1. x = (6 + 2√6) / 2 or x = (6 − 2√6) / 2; 2. x = (6 + 2√15) / 2 or x = (6 − 2√15) / 2; 3. x = (1 + √13) / 2 or x = (1 − √13) / 2; 4. x = (6 + 2√6) / 2 or x = (6 − 2√6) / 2; 5. x = (3 + √33) / 2 or x = (3 − √33) / 2; 6. x = (-5 + √5) / 2 or x = (-5 − √5) / 2; 7. x = (7 + √41) / 2 or x = (7 − √41) / 2; 8. x = (-7 + √61) / 2 or x = (-7 − √61) / 2; 9. x = (2 + 4√2) / 2 or x = (2 − 4√2) / 2; 10. x = (-4 + 2√5) / 2 or x = (-4 − 2√5) / 2
Result 10-problem worksheet — Quadratic equations (hard) (seed 20260820)
No amount of searching will factor these over the integers, which is the point. Time spent looking for a factorisation is time wasted, and recognising that quickly is part of the skill.
The roots come out in surd form, with any square factor extracted from the radical. Leaving them as decimals loses exactness, which is why the key gives the surd.
Reading the result
The level tells you the method
On easy and medium a factorisation exists and is the fast route. On hard none exists, so reaching for the formula immediately is correct rather than lazy.
A non-square discriminant means irrational roots
The roots are rational exactly when the discriminant is a perfect square. Computing it first is therefore a two-second test for whether factoring can possibly work.
Surd form is exact, decimals are not
A root written with a radical is the precise value; the same root as a decimal is rounded. On hard the key uses surds for that reason, and a decimal answer will not match it.
When you would use this
Drilling factoring before the formula
Easy and medium build the recognition that makes factoring fast, which is worth having before the general method removes the need for it.
Practising the formula in surd form
Hard exercises the whole chain: computing the discriminant, simplifying the radical, and reducing the fraction. Sheets where the roots happen to be rational never touch the last two.
Assumptions and limitations
What this calculator assumes
- Easy and medium construct the quadratic from two distinct non-zero integer roots, so both always factor.
- Hard requires a positive, non-square discriminant, so its roots are real and irrational.
- All coefficients are integers on every level.
- The same seed, count and difficulty reproduce the same sheet exactly.
Where it stops being the right tool
- No complex roots: hard insists on a positive discriminant, so the negative case never appears.
- No repeated roots, since the two roots are always drawn distinct.
- Completing the square is not exercised as a separate method.
- One difficulty per sheet, at most 30 problems.
Common mistakes
Hunting for a factorisation on a hard sheet
Why it happens. Every earlier level rewarded that search, and nothing about the equation announces that no integer factorisation exists.
How to avoid it. Compute the discriminant first. If it is not a perfect square, factoring over the integers is impossible and the formula is the only route.
Reporting hard roots as decimals
Why it happens. A decimal looks like a finished answer and is easier to write than a radical, and a calculator produces one readily.
How to avoid it. Leave the radical in place and simplify it. The key gives surds, and a rounded decimal is a different — less exact — answer.
Distributing the leading coefficient on medium
Why it happens. The larger numbers make the quadratic look like a genuinely non-monic factoring problem rather than a scaled monic one.
How to avoid it. Divide the whole equation through by the common factor first. Since the right-hand side is zero, that changes nothing about the roots.
Frequently asked questions
Why integer roots on easy?
Because easy and medium target the factoring stage, where whole-number roots make the patterns visible and let a student verify an answer by substitution. Hard moves on to the general formula, where that shortcut is deliberately removed.
How are the irrational roots reported?
In surd form, as the formula produces them, with any perfect-square factor pulled out of the radical. That keeps the answer exact rather than rounded, which is what an exam expects.
Can the generator make complex-root quadratics?
No. The hard level requires a positive discriminant, so the roots stay real. The aim is practice with the formula and with surds, not with complex arithmetic.
How does the hard level guarantee the formula is needed?
By rejecting any draw whose discriminant is negative or a perfect square, retrying up to thirty times. A non-square discriminant makes the roots irrational, so no factorisation over the integers exists.