Diamond problems are warm-up exercises for factoring quadratics: to factor x² + Sx + P you need two numbers whose product is P and whose sum is S – exactly the diamond. Solving them builds number sense for the ac method.
How to use the Diamond Problem Calculator
- I know…
- Factor A
- Factor B
- Product (top)
- Sum (bottom)
Results update as you type, and everything is calculated in your browser.
How it works
With both factors known, multiply and add them. With the product and sum known, the factors are the roots of t² − St + P = 0, found with the quadratic formula. If S² − 4P is negative there are no real factors.
Tips
- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.
Common mistakes to avoid
- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.
Inputs explained
| Field | What to enter |
|---|---|
| I know… | Choose one: Factor A and factor B, The product and the sum |
| Factor A | A number |
| Factor B | A number |
| Product (top) | A number |
| Sum (bottom) | A number |
What each result means
- Product (top) = [Factor A]×[Factor B]
- Sum (bottom) = [Factor A] + [Factor B]
- Factor A = ([Sum (bottom)] + sqrt([Sum (bottom)]^2 – 4×[Product (top)]))÷2
- Factor B = ([Sum (bottom)] – sqrt([Sum (bottom)]^2 – 4×[Product (top)]))÷2
Example values
How the main result changes when Factor A changes and everything else stays at the example values above:
| Factor A | Product (top) |
|---|---|
| -1.5 | -12 |
| -2.25 | -18 |
| -3 | -24 |
| -3.75 | -30 |
| -4.5 | -36 |
| -6 | -48 |
Formula
Product = A × B, Sum = A + BFrom P and S: A, B = (S ± √(S² − 4P)) / 2Worked example
Inputs — I know…: Factor A and factor B; Factor A: -3; Factor B: 8.
Results — Product (top) = -24; Sum (bottom) = 5.
Key terms
- Factor
- A number that multiplies with another to give a product.
Frequently asked questions
What goes where in a diamond problem?
Product on top, sum on the bottom, the two factors on the left and right.
How does it help factoring?
To factor x² + 5x − 24, find factors with product −24 and sum 5: 8 and −3, so (x + 8)(x − 3).
What if the factors are not whole numbers?
The calculator still finds them; they are the roots of t² − St + P = 0.
What if there are no real factors?
When S² − 4P < 0 no real numbers have that product and sum.
What formula does the Diamond Problem Calculator use?
Product = A × B, Sum = A + B.