# Diamond Problem Calculator

URL: https://speedycalculator.com/calculators/math/diamond-problem/
Last updated: 2026-10-30

> The diamond problem calculator fills in an algebra "diamond": the product goes on top, the sum on the bottom and the two factors left and right. Enter the two factors, or the product and the sum, and it finds the rest.

## Inputs

- I know…
- Factor A
- Factor B
- Product (top)
- Sum (bottom)

## Results

- Product (top)
- Sum (bottom)
- Factor A
- Factor B

## Formula

```
Product = A × B,  Sum = A + B
From P and S: A, B = (S ± √(S² − 4P)) / 2
```

## Explanation

Diamond problems are warm-up exercises for factoring quadratics: to factor x² + Sx + P you need two numbers whose product is P and whose sum is S – exactly the diamond. Solving them builds number sense for the ac method.

### How to use the Diamond Problem Calculator

- **I know…**
- **Factor A**
- **Factor B**
- **Product (top)**
- **Sum (bottom)**

Results update as you type, and everything is calculated in your browser.

### How it works

With both factors known, multiply and add them. With the product and sum known, the factors are the roots of t² − St + P = 0, found with the quadratic formula. If S² − 4P is negative there are no real factors.

### Inputs explained

FieldWhat to enterI know…Choose one: Factor A and factor B, The product and the sumFactor AA numberFactor BA numberProduct (top)A numberSum (bottom)A number

### What each result means

- **Product (top)** = [Factor A]×[Factor B]
- **Sum (bottom)** = [Factor A] + [Factor B]
- **Factor A** = ([Sum (bottom)] + sqrt([Sum (bottom)]^2 - 4×[Product (top)]))÷2
- **Factor B** = ([Sum (bottom)] - sqrt([Sum (bottom)]^2 - 4×[Product (top)]))÷2

### Example values

How the main result changes when **Factor A** changes and everything else stays at the example values above:

Factor AProduct (top)-1.5-12-2.25-18-3-24-3.75-30-4.5-36-6-48

## Worked example

Inputs — I know…: Factor A and factor B; Factor A: -3; Factor B: 8.

Results — **Product (top) = -24**; Sum (bottom) = 5.

## Key terms

- **Factor**: A number that multiplies with another to give a product.

## FAQ

**What goes where in a diamond problem?**
Product on top, sum on the bottom, the two factors on the left and right.

**How does it help factoring?**
To factor x² + 5x − 24, find factors with product −24 and sum 5: 8 and −3, so (x + 8)(x − 3).

**What if the factors are not whole numbers?**
The calculator still finds them; they are the roots of t² − St + P = 0.

**What if there are no real factors?**
When S² − 4P < 0 no real numbers have that product and sum.


## 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)