The diamond method makes factoring quadratic trinomials visual and systematic. Learn exactly how it works — with step-by-step examples for every case, including negatives.
The diamond method is a visual technique for factoring quadratic trinomials of the form x² + bx + c. The diamond diagram stores two key values: the constant c (the product) in the top cell, and the coefficient b (the sum) in the bottom cell. Your goal is to find two numbers A and B such that A × B = c and A + B = b.
Once you find A and B, the factored form is simply (x + A)(x + B). The diamond problem solver does exactly this — enter the product and sum, get A and B instantly with every step shown. Also called the factor diamond, this diagram makes the relationship between product and sum concrete and visual.
Five steps from trinomial to factored form — works for every case.
Identify a, b, and c in x² + bx + c. For factoring with the diamond method (a = 1), the top of the diamond gets c and the bottom gets b.
Place c in the top cell (the product A × B) and b in the bottom cell (the sum A + B). The left and right cells are the unknowns you need to find.
Find two numbers A and B where A × B = c AND A + B = b. Apply sign rules first: same signs if product is positive, opposite signs if product is negative.
The factored form is (x + A)(x + B). If A or B is negative, the minus sign is included automatically — e.g. (x − 2)(x − 3) when A = −2 and B = −3.
Expand (x + A)(x + B) using FOIL to confirm you recover the original trinomial. If the product and sum match, your factoring is correct.
Three worked examples — positive, negative sum, and negative product.
Diamond: top = 12, bottom = 7 → find A = 3, B = 4
Both factors negative (positive product, negative sum) → A = −2, B = −3
Opposite signs (negative product) → larger positive: A = 4, B = −3
Also known as the X method calculator technique — compare all approaches.
| Method | Best For | Requires |
|---|---|---|
| Diamond method | x² + bx + c (a = 1) | Finding a factor pair |
| AC method | ax² + bx + c (a ≠ 1) | Multiply a · c first |
| Quadratic formula | Any trinomial | Works when no integer factors exist |
| FOIL (reverse) | Simple cases | Pattern recognition |