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Diamond Problem Solver
Diamond problem calculator for the x game: find the two numbers with a given product and sum, the factor pair used for the box method and for factoring.
Enter any two numbers. The usual problem puts the product on top and the sum on the bottom.
The numbers are 4 and 9.
Either order works. The smaller number is on the left.
4 × 9 = 36 and 4 + 9 = 13
This diamond problem solver is for algebra class and homework that uses a diamond calculator, the box method, or factoring. The top number is a product and the bottom number is a sum. The left and right numbers multiply to the top and add to the bottom, the same pair used to factor x² + bx + c. Teachers sometimes call that puzzle the x game. Enter the product and the sum, both side numbers, or one side with either total. Product 36 and sum 13 give 4 and 9, because 4 × 9 = 36 and 4 + 9 = 13.
Product on top and sum on the bottom
A diamond problem puts four numbers on a diamond, or sometimes on a large X. The top number is a product, the bottom number is a sum, and the left and right numbers are the pair that multiplies to the top and adds to the bottom. For product 36 and sum 13, list the factor pairs of 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6. The pair that adds to 13 is 4 and 9, because 4 × 9 = 36 and 4 + 9 = 13. Either order works on the sides. The smaller number is placed on the left here.
That pair is the one used to factor x² + bx + c. Here b is 13 and c is 36, so x² + 13x + 36 = (x + 4)(x + 9). The box method uses the same two numbers to split the box, and the x game asks for them on the sides of an X. Diamond problems, the diamond method, and the box method are all looking for this factor pair.
Filling the diamond from any two numbers
If the left and right numbers are the ones already filled in, multiply them for the top and add them for the bottom. Left 4 and right 9 give product 36 and sum 13. If one side and the sum are filled, subtract that side from the sum to get the other side, then multiply. If one side and the product are filled, divide the product by that side. When 4 is one side and the product is 10, the other side is 5/2, which is not an integer. The page shows that fraction, and a decimal only when it is exact or when it is rounded and marked ≈.
A product of 0 means one of the numbers is 0 and the other is the sum, so product 0 and sum 5 give 0 and 5. Negatives follow the same two rules. Product −36 and sum 5 give −4 and 9, because (−4) × 9 = −36 and (−4) + 9 = 5. Those are the numbers for factoring x² + 5x − 36. A positive product and a negative sum means both numbers are negative: product 36 and sum −13 give −9 and −4.
When no integer pair exists
Product 12 and sum 5 have no integer pair. The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4, and the same pairs with both signs flipped. None of those pairs add to 5. No real numbers work either, because 5² − 4 × 12 = −23 is negative. The page says that and stops. It does not invent complex numbers. If the real numbers exist but are not integers, they are shown only as rounded decimals marked ≈. For an algebra class, the useful conclusion is that x² + 5x + 12 does not factor over the integers.
Frequently Asked Questions
How do you solve a diamond with product 36 and sum 13?
List the factor pairs of 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6. Only 4 and 9 add to 13. The check is 4 × 9 = 36 and 4 + 9 = 13. Either order can go on the left and right. Those numbers factor x² + 13x + 36 as (x + 4)(x + 9).
What if no two integers work, such as product 12 and sum 5?
Say that no integer pair has product 12 and sum 5. The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4, along with both numbers negative, and none of them add to 5. No real numbers work either, because 5² − 4 × 12 = −23 is negative. Leave the sides blank rather than inventing complex numbers.
What is the box method, the x game, and the diamond method?
They are other names for this same factor pair. The diamond method draws a diamond, the x game draws a large X, and the box method uses the pair to fill a factoring box. In each one, two numbers must multiply to the product and add to the sum. For x² + bx + c, that product is c and that sum is b.
How do negative numbers work on the diamond?
A negative product means one number is negative and the other is positive. Product −36 and sum 5 give −4 and 9, since (−4) × 9 = −36 and (−4) + 9 = 5. A positive product with a negative sum means both numbers are negative. Product 36 and sum −13 give −9 and −4. A typed hyphen and the minus sign − both count as negative.
What if the product is 0, or only one side is missing?
A product of 0 means one number is 0 and the other equals the sum. Product 0 and sum 5 give 0 and 5. If both sides are filled, multiply for the product and add for the sum. If one side and the sum are filled, subtract. If one side and the product are filled, divide. When that division is not a whole number, the missing side is a fraction such as 5/2, not an integer.