Diamond Problem Calculator (two numbers from their product and sum)

Understanding the diamond problem calculator — concept and principle

This calculator solves diamond problems: find two numbers from their product (top) and sum (bottom), or find the product and sum of two given numbers. It also shows the discriminant and how the answer factors a trinomial. Use it for pre-algebra practice and as the first step of factoring x² + bx + c.

Find two numbers that multiply to the top and add to the bottom.

How it is calculated

The side numbers m and n satisfy m × n = product and m + n = sum.

They are the roots of t² − (sum)·t + (product) = 0:

m, n = (sum ± √(sum² − 4 × product)) ÷ 2

If the discriminant sum² − 4 × product is negative, no real numbers work.
For factoring: x² + bx + c = (x + m)(x + n) when m × n = c and m + n = b.

The principle

Two numbers are completely fixed by their sum and product, because they are the two solutions of one quadratic equation. Expanding (t − m)(t − n) gives t² − (m + n)t + mn, so knowing the sum and product is the same as knowing that quadratic. In factoring, the same pair turns x² + bx + c into a product of two brackets. Whole-number answers exist only when the discriminant is a perfect square.

Worked example

Default inputs: top (product) −24, bottom (sum) 5.

Discriminant = 5² − 4 × (−24) = 25 + 96 = 121 = 11², a perfect square.
m, n = (5 ± 11) ÷ 2 = 8 and −3.
Side numbers: 8, −3; check 8 × (−3) = −24 and 8 + (−3) = 5.
Exact form 8, −3; use in factoring: x² + 5x − 24 = (x + 8)(x − 3).

How to read the result

A negative product means the two numbers have opposite signs; a positive product with a negative sum means both are negative. If the discriminant is not a perfect square, the numbers are irrational and the trinomial does not factor over the whole numbers, though the quadratic formula still solves it.

Common mistakes

Swapping the roles of top and bottom.
Finding a pair with the right product but the wrong sign for the sum.
Stopping at whole-number guesses when the answer is irrational.
Forgetting that the factors are (x + m)(x + n), using the same signs as m and n.

Keep in mind: Diamond problems with a negative discriminant have no real solution; the page reports this instead of guessing.

Frequently asked questions

How do you solve a diamond problem?

Find two numbers whose product is the top and whose sum is the bottom.

How does it help with factoring?

For x² + bx + c, the diamond with top c and bottom b gives the numbers in (x + m)(x + n).

What if no whole numbers work?

Use the formula with the square root; the numbers may be irrational or not real.

Sources: OpenStax, Elementary Algebra 2e (2020), §7.2 Factor Trinomials of the Form x² + bx + c

Checked 9 Nov 2026. OpenStax, Elementary Algebra 2e (2020), §7.2 Factor Trinomials of the Form x² + bx + c (find two numbers m and n with m·n = c and m + n = b; then x² + bx + c = (x + m)(x + n)).

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