Calculate the two numbers in a diamond problem from the product and sum, and display the matching left and right values.
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Diamond Problem Formula
A diamond problem uses two numbers that multiply to a product and add to a sum. If the product is P and the sum is S, the two missing numbers can be found with the quadratic formula.
- P = the product, or the number at the top of the diamond
- S = the sum, or the number at the bottom of the diamond
- x = the left missing number
- y = the right missing number
- S^2 - 4P = the discriminant, which tells whether a real-number answer exists
The calculator takes the product and sum you enter, then solves for the two real numbers that satisfy both equations. It uses a stable equivalent of the quadratic formula to reduce cancellation; displayed decimals are rounded. The left and right values may appear in either order because multiplication and addition are commutative. For example, 3 and 5 give the same product and sum as 5 and 3.
Diamond Problem Result Checks
| Discriminant | Meaning | Result type |
|---|---|---|
| S2 - 4P > 0 | There are two different real numbers. | Example: 2 and 6 |
| S2 - 4P = 0 | Both missing numbers are the same. | Example: 4 and 4 |
| S2 - 4P < 0 | No real-number pair matches the product and sum. | No real answer |
| Product | Sum | Missing numbers |
|---|---|---|
| 12 | 7 | 3 and 4 |
| 18 | 9 | 3 and 6 |
| -10 | 3 | -2 and 5 |
| 16 | 8 | 4 and 4 |
Example Diamond Problems
Example 1: Product 24 and sum 10
Find two numbers that multiply to 24 and add to 10.
The missing numbers are 4 and 6.
Example 2: Product -12 and sum 1
Find two numbers that multiply to -12 and add to 1.
The missing numbers are -3 and 4.
FAQ
What is a diamond problem?
A diamond problem is a math puzzle where you use a product and a sum to find two missing numbers. The two missing numbers must multiply to the product and add to the sum.
Can a diamond problem have negative numbers?
Yes. If the product is negative, one missing number is positive and the other is negative. For example, a product of -10 and a sum of 3 gives -2 and 5.
Why does the calculator say there is no real-number pair?
This happens when the discriminant, S2 - 4P, is negative. In that case, no two real numbers can have both the product and sum you entered. The calculator only returns real-number answers.
