How to use the factoring quadratics worksheet generator
- Enter the number of problems, from 1 to 30, and a worksheet number.
- Choose the leading coefficient: always 1, or can be 2 or 3 for harder problems.
- Tick Show the answer key to see each quadratic written as a product of two binomials.
- Print the sheet for students, and keep the key for yourself.
Formula
With a leading coefficient of 1: x^2 + bx + c = (x + p)(x + q), where p + q = b and p x q = c
In general: (ax + b)(cx + d) = ac x^2 + (ad + bc) x + bd
Factors are whole numbers, and every sheet problem factors over the integers
Check by multiplying the factors back out with the distributive property
For a leading coefficient of 1, find two whole numbers that multiply to the constant and add to the middle coefficient. For other leading coefficients, try the possible pairs for a and c and check the middle term.
The first problems of worksheet number 1
Problem 1 is x^2 - 7x - 18. The numbers 2 and -9 multiply to -18 and add to -7, so the factors are (x + 2)(x - 9). Problem 2 is x^2 + 10x + 9, and 1 and 9 give (x + 1)(x + 9).
Problem 3 is x^2 + 5x - 36, and 9 and -4 multiply to -36 and add to 5, so the factors are (x + 9)(x - 4). Problem 4 is x^2 + 6x + 8, which factors as (x + 2)(x + 4). Check the first by expanding: (x + 2)(x - 9) = x^2 - 9x + 2x - 18 = x^2 - 7x - 18.
Building the skill
Start with a leading coefficient of 1, where students list the factor pairs of the constant and look for the pair with the right sum. Signs matter: a positive constant means both numbers have the same sign, and a negative constant means the signs are opposite. Once that is secure, turn on the harder option, where the leading coefficient is 2 or 3 and students must test combinations.
To check a factorization, solve the equation with the quadratic equation solver. If the roots are -2 and 9, the factors are (x + 2) and (x - 9).
Mistakes to look for
Sign errors are the most common problem, followed by choosing a pair that has the right product but the wrong sum. Some students forget to check the middle term, and some leave a common factor outside. Ask students to multiply back every time, since the check takes only seconds.
For related practice, see the exponent rules worksheet generator and the GCF and LCM calculator, which finds a common factor to take out first. The log, exponent and root calculator helps with the roots.
Assumptions and limits
- Every problem factors into binomials with whole number coefficients.
- The constants in the binomials are whole numbers from -9 to 9, and are never zero.
- In the harder option a quadratic can have a common factor, so remind students to look for one first.
- The sheet is general practice and is not matched to a particular curriculum.
Frequently asked questions
How do you factor a quadratic?
For x^2 + bx + c, find two numbers that multiply to c and add to b, then write (x + p)(x + q). If the leading coefficient is not 1, test pairs of factors and check the middle term.
How can I check my factoring?
Multiply the binomials back out. You should get the original quadratic. Or solve the equation and compare the roots.
What does the harder option do?
It lets the leading coefficient of each binomial be 1, 2 or 3, so the quadratic can have a leading coefficient up to 9.
Can I make a different sheet?
Yes. Change the worksheet number. The same number and options always make the same sheet.