How to Factor Quadratics Step by Step
By the BrainSnail editorial team. How these articles are written and checked, and how to tell us when one is wrong.
Factoring quadratics is easier when you work backward from multiplication. You are looking for two brackets whose product expands to the original quadratic expression.
How to factor quadratics when the first coefficient is 1
Start with a quadratic such as x^2 + 7x + 12. You want two numbers that multiply to 12 and add to 7. The pair 3 and 4 works, so the factorization is (x + 3)(x + 4). Expanding the brackets confirms the result: x^2 + 4x + 3x + 12 becomes x^2 + 7x + 12.
For x^2 - x - 12, look for two numbers that multiply to -12 and add to -1. The numbers -4 and 3 work, giving (x - 4)(x + 3). A negative product means the two numbers have opposite signs.
This is the quickest version of factoring quadratics. Find the product from the constant term, find the sum from the coefficient of x, and place the two numbers into brackets with x.
Use the product-sum method when the first coefficient is not 1
Now consider 2x^2 + 7x + 3. Multiply the first coefficient and the constant: 2 times 3 equals 6. Find two numbers that multiply to 6 and add to 7. Those numbers are 6 and 1.
Split the middle term using that pair: 2x^2 + 6x + x + 3. Group the expression as 2x(x + 3) + 1(x + 3). The matching bracket appears twice, so factor it out to get (2x + 1)(x + 3).
When factoring quadratics, write every stage rather than trying to jump straight to the brackets. The split-middle-term method is slower on paper but reliable, especially when signs or larger coefficients make mental guessing harder.
Check signs and know when factoring will not be simple
Always expand your answer. Multiply the first terms, outer terms, inner terms, and last terms, then combine like terms. If you recover the original quadratic exactly, the factorization is correct. This check takes only a few seconds and catches swapped signs or missing coefficients.
Some quadratics do not factor neatly using integers. For example, a quadratic may have irrational or complex roots, so no pair of simple integer brackets will work. In an equation, you may need the quadratic formula or another method instead. The instructions in a school problem usually hint whether simple factoring is expected.
A special pattern can also save time. A difference of two squares, such as x^2 - 25, factors as (x - 5)(x + 5). If every term has a common factor, take that out first. Good how to factor quadratics habits begin with a quick scan for common factors and familiar patterns before using the full product-sum method. Factoring is especially useful when a quadratic expression is set equal to zero. Once you have (x + 3)(x + 4) = 0, the zero-product rule says at least one bracket must equal zero. That gives x = -3 or x = -4. The factorization therefore reveals the roots directly, which is one reason the skill matters beyond simplifying algebraic expressions. If you get stuck, list factor pairs systematically instead of guessing at random. For a positive constant, test pairs with matching signs. For a negative constant, test opposite signs. Then use the middle coefficient to decide which pair has the required sum. This keeps the search small and organised.
The takeaway
Factoring quadratics means reversing expansion. For a leading coefficient of 1, find two numbers with the right product and sum. For other leading coefficients, multiply the first and last coefficients, split the middle term, and factor by grouping. Expand your final brackets to check the answer.