# Generic Rectangle Calculator

URL: https://speedycalculator.com/calculators/math/generic-rectangle/
Last updated: 2026-10-30

> The generic rectangle calculator factors ax² + bx + c with an area model: the four cells hold ax², two x-terms and c, and the side lengths of the rectangle are the binomial factors.

## Inputs

- a (x² coefficient)
- b (x coefficient)
- c (constant)

## Results

- Factored form
- Generic rectangle
- Steps

## Formula

```
Cells: ax², rx, sx, c with r·s = a·c and r + s = b
```

## Explanation

A generic rectangle is an area model of multiplication. Because the area of a rectangle is length × width, a trinomial written as the area of a 2 × 2 rectangle reveals its factors on the edges. It works for multiplying binomials too.

### How to use the Generic Rectangle Calculator

- **a (x² coefficient)**
- **b (x coefficient)**
- **c (constant)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The calculator finds the split of the middle term, fills the rectangle and divides out the greatest common factor of each row and column to read off the dimensions.

## Worked example

Inputs — a (x² coefficient): 2; b (x coefficient): 7; c (constant): 3.

Results — **Factored form = (2x + 1)(x + 3)**; Generic rectangle = × │ x │ 3
2x │ 2x² │ 6x
1 │ 1x │ 3; Steps = 1. Find two numbers whose product is a·c = 6 and whose sum is b = 7
2. They are 1 and 6. Split the middle term: 2x² + 1x + 6x + 3
3. Group and factor: 1x(2x + 1) + 3(2x + 1)
4. Result: (2x + 1)(x + 3).

## Key terms

- **Area model**: Representing multiplication as the area of a rectangle.

## FAQ

**What is a generic rectangle?**
A rectangle split into parts whose areas are the terms of a polynomial, used to multiply and factor.

**Why do the diagonals multiply to the same value?**
ax² × c and rx × sx both equal acx² – this is why r·s must equal a·c.

**Is it different from the box method?**
It is the same idea with a different name.

**Can I use it for bigger polynomials?**
Yes, with more rows and columns, one for each term.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Intermediate Algebra 2e](https://openstax.org/details/books/intermediate-algebra-2e)