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Diamond Problem Calculator | Find Numbers With Sum & Product

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Solve diamond problems — find two numbers that add to the top value and multiply to the bottom — with the quadratic formula shown. The classic algebra factoring tool for students and teachers.

Last updated: June 2026 · Free · No sign-up required

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Results

Enter values above and click Calculate to see your result instantly.

Quick reference

SumProductThe two numbers
7123 and 4
562 and 3
−1−123 and −4
10255 and 5

info Diamond Problem Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

Discriminant s² − 4pResult
PositiveTwo distinct real numbers
ZeroOne repeated number (s/2 twice)
NegativeNo real solution — the pair is complex

lightbulb Worked example — sum 11, product 24

A student is factoring and needs two numbers that add to 11 and multiply to 24. The numbers are the roots of x² − 11x + 24 = 0.

Result: Numbers = [11 ± √(11² − 4×24)] ÷ 2 = [11 ± √(121 − 96)] ÷ 2 = [11 ± √25] ÷ 2 = [11 ± 5] ÷ 2, giving 16 ÷ 2 = 8 and 6 ÷ 2 = 3. Check: 8 + 3 = 11 and 8 × 3 = 24.

What this means: The pair is 8 and 3, so x² − 11x + 24 factors as (x − 8)(x − 3). The discriminant (25) was a perfect square, which is why the pair came out as whole numbers — most classroom diamond problems are designed this way, but the formula works just as well when it isn’t.

Formula & methodology

Formula: Numbers = [s ± √(s² − 4p)] / 2 — the roots of x² − sx + p = 0

How a diamond problem is solved

Two numbers with sum s and product p are the roots of the quadratic x² − sx + p = 0, which the quadratic formula solves directly — no guessing needed.

  • Read the sum s (bottom of the diamond) and the product p (top).
  • Compute the discriminant: s² − 4p.
  • Apply the formula: numbers = [s ± √(s² − 4p)] ÷ 2.
  • Verify by adding and multiplying the pair back.

Reading the discriminant: a positive value gives two distinct real numbers, zero gives one repeated number (s/2 twice), and a negative value means no real pair exists — the solutions are complex. Example: sum 7, product 12 → [7 ± √(49 − 48)] ÷ 2 = 4 and 3.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is the diamond problem in math?
A diamond (X) diagram with a product on top and a sum on the bottom: you find the two side numbers that multiply to the top and add to the bottom. It's the core skill for factoring quadratics.
How do you solve a diamond problem?
Enter the required values into the input fields and press the calculate button. The tool applies the formula Numbers = [s ± √(s² − 4p)] / 2 — the roots of x² − sx + p = 0 and returns the result with a short explanation of each step, so you can check the working yourself.
What two numbers add to 7 and multiply to 12?
The Diamond Problem Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Solves diamond (X) problems used to practise factoring quadratics. Enter the two numbers that must add to the top value and multiply to the bottom value, and the tool finds the missing pair — the two numbers that satisfy both the sum and the product — with There is no limit on usage, and it works on any device.
What if no whole numbers work?
The two numbers can be irrational or complex: the formula [s ± √(s² − 4p)] ÷ 2 always produces valid values, but the square root may not be a whole number. When s² − 4p is negative, the pair involves complex numbers, which means the quadratic has no real factorization — the diamond puzzle has no real solution. The calculator shows the exact values either way.
How does the diamond method help factor trinomials?
The Diamond Problem Calculator uses the standard Numbers = [s ± √(s² − 4p)] / 2 — the roots of x² − sx + p = 0. You enter your values, and the calculator applies the formula step by step, showing the working so you can verify the math and understand how the result is derived rather than trusting a black box.
How does the diamond diagram connect to factoring quadratics?
Factoring x² + bx + c means finding two numbers that multiply to c and add to b — exactly what the diamond asks for. For x² + 7x + 12, the diamond with top 12 and bottom 7 gives 3 and 4, so the trinomial factors as (x + 3)(x + 4). The diamond is just a visual organiser for the same pair of numbers.
What if the discriminant is not a perfect square?
The formula still works — you just get irrational numbers instead of whole ones. With sum 5 and product 6, the discriminant is 25 − 24 = 1 and the pair is 3 and 2; but with sum 5 and product 7, the discriminant is −3, so the pair is complex: (5 ± i√3)/2. Classroom problems usually stick to perfect squares.
Why are the two numbers unordered — does 3-and-4 differ from 4-and-3?
No — addition and multiplication are both commutative, so the pair is the same set either way. The ± in the formula produces both numbers at once, and swapping their labels changes nothing about the factorisation or the diamond.
Can a diamond problem have no real-number solution?
Yes. When s² − 4p is negative, no two real numbers can satisfy both the sum and the product — the pair must be complex. For example, sum 2 and product 5 gives a discriminant of 4 − 20 = −16, so the numbers are 1 ± 2i. Such a diamond would never appear in a whole-number factoring exercise.

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