Step 1 — Read the Diamond Correctly
Before solving, confirm which values are given and where they appear in the diamond. The layout is always the same regardless of which textbook or teacher you have:
- Top cell — the product (A × B). The two side numbers multiply to give this value.
- Bottom cell — the sum (A + B). The two side numbers add to give this value.
- Left and right cells — the two unknown numbers A and B that you need to find.
💡 Memory tip: "Product on top, sum on the bottom." The two numbers you're finding go on the sides.
Step 2 — Apply Sign Rules Before Listing Factor Pairs
This is the step most students skip — and the source of the majority of diamond problem errors. The signs of the product and sum tell you the signs of A and B before you list a single pair. Apply the rule first, then list factor pairs.
| Product | Sum | Both Factors Are… | Why |
|---|---|---|---|
| Positive + | Positive + | Both positive | Two positive numbers multiply positively and add positively |
| Positive + | Negative − | Both negative | Two negatives multiply to a positive, but add to a negative |
| Negative − | Positive + | Opposite signs; positive is larger | Opposite signs multiply to negative; larger positive makes sum positive |
| Negative − | Negative − | Opposite signs; negative is larger | Opposite signs multiply to negative; larger absolute negative makes sum negative |
Testing only positive factor pairs when the product is positive but the sum is negative. Positive product + negative sum means BOTH factors are negative. Apply the sign rule first — always.
Step 3 — List Factor Pairs and Find the Match
Once you know the signs, list all factor pairs of the absolute value of the product. Apply the sign from step 2 to each pair. Then check which pair's sum matches the target sum.
Sign rule: positive product + positive sum → both factors positive.
Factor pairs of 12: (1, 12) → sum 13 ✗ · (2, 6) → sum 8 ✗ · (3, 4) → sum 7 ✓
Answer: A = 3, B = 4
Sign rule: negative product → opposite signs. Positive sum → positive factor has larger absolute value.
Factor pairs of 20 with opposite signs: (−1, 20) → 19 ✗ · (−2, 10) → 8 ✗ · (−4, 5) → 1 ✓
Answer: A = 5, B = −4
Step 4 — Verify Both Conditions
Before writing the final answer, confirm that both conditions hold simultaneously. This catches sign errors and arithmetic mistakes:
- Check: A × B = product (top) ✓
- Check: A + B = sum (bottom) ✓
If either check fails, revisit your sign rules or factor pair list. A single sign error is the most common cause of a failed verification.
Quadratic Formula — For Non-Integer Answers
When no integer pair satisfies both conditions, use the quadratic formula. Given product P and sum S, the two numbers x and y satisfy:
Discriminant = 4² − 4×5 = 16 − 20 = −4. Negative discriminant → no real solution.
x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2
x = 3 or x = 2. Verify: 3 × 2 = 6 ✓, 3 + 2 = 5 ✓
Answer: A = 3, B = 2
When There Is No Solution
If the discriminant (S² − 4P) is negative, the diamond problem has no real number solution. Your teacher may intentionally include such problems to test whether students recognize them. In a factoring context, a no-solution diamond problem means the quadratic cannot be factored over the integers and requires the quadratic formula.
How Diamond Problems Connect to Factoring
Every time you factor x² + bx + c, you are solving a diamond problem. The product is c (constant term) and the sum is b (x-coefficient). The two numbers you find become the constants in the factored form (x + A)(x + B).
Diamond: product = 6, sum = −5
Sign rule: positive product + negative sum → both factors negative
Factor pairs of 6: (−1,−6)→−7 ✗ · (−2,−3)→−5 ✓
x² − 5x + 6 = (x − 2)(x − 3)
Check Your Answer Instantly
Use the diamond problem solver to verify your work or get a full step-by-step solution.
◆ Open Solver →