A factor diamond is the visual tool used in the diamond method to factor quadratic trinomials. Learn what it is, how it works, and use the free solver to check your answers.
A factor diamond is a diamond-shaped diagram with four cells. The top cell holds the product of two unknown numbers, the bottom cell holds their sum, and the goal is to find the two numbers that go in the left and right cells. It is a purely visual tool that makes the product-sum relationship explicit and easy to work with.
It is the central tool of the diamond method for factoring trinomials. When factoring x² + bx + c, the top cell gets c (the product) and the bottom cell gets b (the sum). Solving the factor diamond gives you the two numbers A and B that complete the factored form (x + A)(x + B). Also known as the X method calculator technique.
Four steps from trinomial to fully factored form.
Write the trinomial in the form x² + bx + c and note the values of b (x-coefficient) and c (constant term). These are the two values you'll place in the factor diamond.
Place c in the top cell (product) and b in the bottom cell (sum). The left and right cells are empty — those are the unknowns A and B you need to find.
Find two numbers where A × B = c and A + B = b. Apply sign rules first: if c is positive, A and B have the same sign (both follow the sign of b). If c is negative, A and B have opposite signs.
Write the answer as (x + A)(x + B). Expand using FOIL to verify you get the original trinomial back. The diamond problem solver checks this instantly.
Two worked examples with fully labelled factor diamonds.
c = 12, b = 7 → factor pairs: (1,12), (2,6), (3,4) → 3 + 4 = 7 ✓
c = 6, b = −5 → both negative (same sign, follows b) → −2 × −3 = 6, −2 + (−3) = −5 ✓