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Quadratic Equations Worksheet Generator

Printable quadratic practice for every method, with exact answers – whole numbers, fractions or simplified radicals – in the answer key.

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This quadratic equations worksheet generator makes printable practice sheets for solving quadratic equations, with an answer key. It runs in your web browser, is free, and needs no sign-up. You choose the problem types, difficulty and number of problems, and the sheet is made on your own device, so nothing is uploaded.

The generator covers six methods: solving by square roots, solving by factoring, completing the square, the quadratic formula, the discriminant, and the zero product property. You can use it for homework, class practice, or a quiz, and you can make as many different versions as you need. The guide below explains how to use the tool and the methods behind the problems. Everything you make here is free to use in class or in products you sell, such as Teachers Pay Teachers resources, Etsy printables or KDP workbooks.

Quadratic equations worksheet example

How to use the Quadratic Equations Worksheet Generator

  1. Tick one or more boxes under Problem types. The Solve by factoring box is ticked when the page opens.
  2. Choose a level under Difficulty: Easy, Medium or Hard.
  3. Pick how many problems you want under Number of problems, from 6 to 30.
  4. Open More options to set the Columns (Automatic, 1, 2 or 3), the Paper size (Letter or A4), a Worksheet title, and whether to add a Name and date line. Leave Answer key on a separate page ticked if you want the solutions printed after the worksheet.
  5. Check the preview on the right. It shows the sheet as it will print.
  6. Click New worksheet for a fresh set of problems with the same settings. Click Show answers to see each answer under its problem in the preview, and click again to hide them.
  7. Click Print / Save as PDF. In the print dialog, choose Save as PDF if you want a file instead of paper.

When you tick several problem types, the problems are shared evenly between them. Each type gets its own one-line instruction, and the problems are numbered 1 to N across the whole sheet. The settings are remembered in this browser for your next visit.

How do you solve a quadratic equation by factoring

Write the equation in standard form, ax² + bx + c = 0, then factor the left side. Set each factor equal to zero and solve the resulting simple equations. This works because of the zero product property: if A · B = 0, then A = 0 or B = 0.

For example, take x² − 8x + 12 = 0. It factors as (x − 2)(x − 6) = 0, so x − 2 = 0 or x − 6 = 0, which gives x = 2 or x = 6. Some problems need rearranging first. The equation x² + 48 = −14x becomes x² + 14x + 48 = 0, which factors as (x + 6)(x + 8) = 0, giving x = −6 or x = −8. A quadratic equation has at most two real solutions, so you should expect two factors to give two answers.

What is the quadratic formula

The quadratic formula gives the solutions of any quadratic equation ax² + bx + c = 0, where a ≠ 0: x = (−b ± √(b² − 4ac)) / (2a). It is useful when the equation does not factor neatly, because it always works.

For example, take 2x² − 9x + 3 = 0, where a = 2, b = −9 and c = 3. Then b² − 4ac = 81 − 24 = 57, and the solutions are x = (9 ± √57) / 4. Write the answer in this exact form, since the root does not simplify to a whole number.

How do you solve by taking square roots?

When there is no x term, get the squared part alone and take the square root of both sides, remembering both the positive and the negative root. For x² = 121, x = ±11. For (x + 8)² = 9, x + 8 = 3 or x + 8 = −3, so x = −5 or x = −11. If the number is not a perfect square, the answer stays exact: (x − 6)² = 57 gives x = 6 ± √57. The square-root problems on the Hard level produce answers like this, and the key always simplifies the radical.

Whichever method you use, the solutions can be checked by substituting each one back into the original equation; both sides must come out equal.

Sample problems from this generator

  1. x2 − 6 = x Answer: x = −2, x = 3
  2. x2 + 49 = 14x Answer: x = 7
  3. x2 = −2x + 3 Answer: x = −3, x = 1
  4. x2 + 9x = −14 Answer: x = −7, x = −2

Frequently asked questions

What does the discriminant tell you?

The discriminant is b² − 4ac, and its value shows how many real solutions the equation has. A positive value gives two real solutions, zero gives one real solution (a double root, written once), and a negative value gives no real solutions. For 3x² + 8x + 9 = 0, the discriminant is 64 − 108 = −44, so there are no real solutions.

How do you complete the square?

Divide by a first if the x² term has a coefficient other than 1. Then add (b/2)² to both sides, write the left side as (x + b/2)², and take square roots. For 3x² + 36x + 42 = 0, dividing by 3 gives x² + 12x + 14 = 0, then (x + 6)² = 22, so x = −6 ± √22.

Is there an answer key?

Yes. The Answer key on a separate page checkbox under More options is on by default. When you print, the key starts on a new page after the worksheet, lists every problem number with its exact answer: whole numbers, reduced fractions or simplified radicals.

Can I get the same worksheet again?

Yes. Click Copy link to copy a link that opens exactly the same worksheet, because the problem set is stored in the link as a number. Anyone who opens the link sees that worksheet, using the settings saved in that link.

Does it work on a phone or tablet?

Yes, it works on phones and tablets as well as computers. You still need a printer or the Save as PDF option to make a printed or digital copy.

Can I sell these quadratic equations worksheets as a printable on Etsy or TpT?

Yes. The worksheets and answer keys you generate are free for personal, classroom and commercial use, such as Teachers Pay Teachers products, Etsy listings or a printed workbook. You do not need to credit this site. You may not copy the generator itself and offer it as your own tool.