Enter the product and sum. Get both missing numbers with a full step-by-step solution. No login, no limits.
Step 2 β Apply the quadratic formula
t = (7 Β± β(49 β 48)) / 2 = (7 Β± β1) / 2 = (7 Β± 1) / 2
Step 3 β Calculate both values
x = (7 + 1) / 2 = 4
y = (7 β 1) / 2 = 3
β Verify: 3 + 4 = 7 β and 3 Γ 4 = 12 β
Type the number that goes in the top cell (product) and the bottom cell (sum) of your diamond problem.
The calculator sets up x + y = sum and x Γ y = product, then applies the quadratic formula to find both values.
See x and y displayed in the diamond, plus a verified step-by-step breakdown showing exactly how each answer was found.
Stuck on a diamond problem worksheet? Get the answer and understand the method before moving on.
Practice before an algebra test. Use the step-by-step solution to confirm your own working is correct.
Diamond problems are the gateway to factoring quadratics. Use the calculator to build pattern recognition.
Teachers can use this to generate worked examples quickly, or let students check their answers independently.
Results shown in an actual diamond shape β product on top, sum on bottom, x and y on the sides. Matches exactly how teachers present it.
Not just the answer β the full working, including equation setup, quadratic formula application, and verification.
Every solution is checked: confirms x + y equals the sum and x Γ y equals the product before showing the result.
Works with any real number input β integers, decimals, negatives. Tells you clearly when no real solution exists.
| Feature | DiamondProblemCalculator | Mathway | Symbolab | CalculatorSoup |
|---|---|---|---|---|
| Diamond-specific interface | β Built for it | β | β | β |
| Visual diamond output | β | β | β | β |
| Step-by-step solution | β Free preview | Paid only | Paid only | β |
| No signup required | β | β | β | β |
| Answer verification | β Always | β | β | β |
A diamond problem is a foundational algebra exercise where students are given a diamond-shaped diagram with four cells β top, bottom, left, and right. The top cell contains the product of two unknown numbers, and the bottom cell contains their sum. The goal is to find the two numbers that go in the left and right cells.
The diamond problem calculator automates this process β but understanding what's happening mathematically is what makes the exercise useful for algebra students in the first place.
Given: top = x Γ y (product) and bottom = x + y (sum). Find: x and y. This is equivalent to solving a quadratic equation, which is exactly what diamond problems are designed to practice.
Diamond problems exist to build one specific skill: the ability to find two numbers that simultaneously satisfy a multiplication condition and an addition condition. That skill is the core of factoring quadratic trinomials.
When students later encounter an expression like xΒ² + 7x + 12 and need to factor it into (x + 3)(x + 4), they need to find two numbers that multiply to 12 and add to 7. Diamond problems train exactly that pattern β so when students hit factoring, the mental process is already familiar.
Top: 12 (product) Β· Bottom: 7 (sum) β Find x and y where x Γ y = 12 and x + y = 7. Answer: 3 and 4.
Factor xΒ² + 7x + 12 β Find two numbers that multiply to 12 and add to 7 β (x + 3)(x + 4). Same problem, different context.
The diamond problem calculator uses the quadratic formula to find x and y from any valid product and sum. Here's the full method:
Not every combination of product and sum has a real number solution. When the discriminant (SΒ² β 4P) is negative, the square root produces an imaginary number β meaning no two real numbers satisfy both conditions simultaneously.
| Product | Sum | Discriminant (SΒ²β4P) | Result |
|---|---|---|---|
| 12 | 7 | 49 β 48 = 1 | β Two real solutions: 3 and 4 |
| 12 | 8 | 64 β 48 = 16 | β Two real solutions: 6 and 2 |
| 12 | 5 | 25 β 48 = β23 | β No real solution |
| β6 | 1 | 1 + 24 = 25 | β Solutions: 3 and β2 |
| 9 | 6 | 36 β 36 = 0 | β One solution: 3 and 3 |
For integer problems, list all factor pairs of the product and check which pair adds up to the sum. For product = 12: pairs are (1,12), (2,6), (3,4). Check sums: 13, 8, 7. If the target sum is 7, the answer is 3 and 4.
When the product is positive and the sum is negative, both numbers are negative. When the product is negative, one number is positive and one is negative. Getting the signs right is where most students lose points on tests.
Solve the problem manually first, then verify with the diamond problem calculator. If your answer doesn't match, the step-by-step solution shows exactly where the method diverged.
"Had a whole worksheet of diamond problems due in 20 minutes. Used this to check my answers and caught two sign errors I would have missed."
"The step-by-step actually showed me what I was doing wrong. I didn't understand why you use the quadratic formula β now I do."
"Good for checking homework. Would be nice to have a practice mode where it generates random problems, but the calculator itself works well."