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Mathematics

Diamond Problem Solver

Solve diamond math problems. Free online Diamond Problem Solver. Calculate diamond problem solver online — fast, accurate, mobile-friendly, no signup needed.

Given product P and sum S, find numbers a, b with a+b=S, ab=P

Number a
4
Number b
3

Derivation

  1. ├── 01GivenP = 12, S = 7
  2. ├── 02FormulaNumber a: {let t=e.P,a=e.S;return(a+√(a²-4 × t)) / 2}
  3. ├── 03Substitute{let t=e.12,a=e.7;return(a+√(a²-4 × t)) / 2}
  4. ├── 04Compute Number a4
  5. ├── 05FormulaNumber b: {let t=e.P,a=e.S;return(a-√(a²-4 × t)) / 2}
  6. ├── 06Substitute{let t=e.12,a=e.7;return(a-√(a²-4 × t)) / 2}
  7. └── 07Compute Number b3
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§01What is

Understanding the Diamond Problem Solver

The Diamond Problem Solver computes Number a from 2 inputs: product, sum. Solve diamond math problems.

Mathematics shows up in every corner of daily life — budgeting, cooking, construction, engineering, even reading a bus schedule. A calculator like this lets you skip the scratch-paper step and move straight to the answer, without the arithmetic mistakes that creep in when the numbers get messy. The Diamond Problem Solver sits in that toolkit — it solve diamond math problems. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

Number a = {let t=e.P,a=e.S;return(a+√(a²-4 × t)) / 2} | Number b = {let t=e.P,a=e.S;return(a-√(a²-4 × t)) / 2}

Where

P
Product
S
Sum
Number a
Output value
Number b
Output value
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Product = 12, Sum = 7.

  1. 01Start by noting the input — Product: 12.
  2. 02Start by noting the input — Sum: 7.
  3. 03Substitute these values into the formula: Number a = {let t=e.P,a=e.S;return(a+√(a²-4 × t)) / 2} | Number b = {let t=e.P,a=e.S;return(a-√(a²-4 × t)) / 2}
  4. 04Compute Number a: the calculator returns 4.
  5. 05Compute Number b: the calculator returns 3.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Diamond Problem Solver Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Product halved

P = 6 (from 12)

Keep every other input at its default and halve the product. See how number a responds.

  1. 01New Product: 6
  2. 02Baseline Number a: 4
  3. 03New Number a: 6
  4. 04Number a increases by 50% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Sum doubled

S = 14 (from 7)

Keep every other input at its default and double the sum. See how number a responds.

  1. 01New Sum: 14
  2. 02Baseline Number a: 4
  3. 03New Number a: 13.0828
  4. 04Number a increases by 227.1% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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