Box Method Calculator: Multiply Binomials With a Grid
This box method calculator multiplies two binomials in a 2×2 grid. Enter (x + 3)(x − 5), and it fills in each cell, combines like terms, and returns x² − 2x − 15.
Use the Box Method Calculator
Type two binomials in parentheses, such as (2x + 1)(3x − 4), and select Calculate. You'll see the grid with each term along the edges, the product in every cell, and the simplified answer.
Input tips
- Write each binomial in parentheses: (2x + 1)(3x - 4).
- Type ^2 or ² for squares, such as (x - 7)² or (2x^2 - 5)(x + 3).
- Two variables work too: (3x + 2y)(x - y).
- Decimals are fine, e.g. (0.5x + 2)(x - 4).
- Select Clear to start a new problem.
What Is the Box Method?
The box method, also called the grid method or area model, multiplies polynomials by placing their terms along the sides of a grid. Each cell holds the product of its row and column. Add the cells and combine like terms to get the answer.
It uses the same distributive property as the FOIL method. The grid just makes it easy to see that every term meets every other term exactly once.
How the Box Method Calculator Works
Here is the grid for (2x + 1)(3x − 4):
| × | 3x | −4 |
|---|---|---|
| 2x | 6x² | −8x |
| 1 | 3x | −4 |
- Set up the grid: 2x and 1 down the side, 3x and −4 across the top.
- Fill each cell: 2x · 3x = 6x², 2x · (−4) = −8x, 1 · 3x = 3x, 1 · (−4) = −4.
- Combine like terms: −8x + 3x = −5x.
- Write the answer: 6x² − 5x − 4.
Keep each sign with its term. The −4 goes in the grid as −4, so every product in its column picks up the minus sign.
Box Method Examples
| Problem | Grid cells | Answer |
|---|---|---|
| (x + 3)(x − 5) | x², −5x, 3x, −15 | x² − 2x − 15 |
| (x − 7)² | x², −7x, −7x, 49 | x² − 14x + 49 |
| (x + 5)(x − 5) | x², −5x, 5x, −25 | x² − 25 (difference of squares) |
| (2x² − 5)(x + 3) | 2x³, 6x², −5x, −15 | 2x³ + 6x² − 5x − 15 |
| (3x + 2y)(x − y) | 3x², −3xy, 2xy, −2y² | 3x² − xy − 2y² |
Box Method vs. FOIL
- Same answer: both apply the distributive property and produce the same four products.
- Fewer sign errors: the grid keeps every term and its sign in view.
- Scales up: a binomial times a trinomial uses a 2×3 grid. FOIL only works for two binomials.
- Works backward: the grid is also a factoring tool. See the reverse foiling calculator.
This calculator handles two binomials (a 2×2 grid). For the same problem in First, Outer, Inner, Last order, use the FOIL calculator.
Try the Box Method Calculator
Put one binomial on each side of the grid, multiply every cell, and add the like terms. Try the box method calculator above with a homework problem, or check the same product with the multiply binomials calculator.
Frequently Asked Questions
Personalized questions and guidance about this specific tool.
Related Calculators
Explore our specialized tools to expand, factor, and master algebraic equations step-by-step.