Diamond Problem Solver

Solve diamond math problems. Enter any two values to find the other two with step-by-step equations shown.

Solve diamond math problems. Find two numbers from their sum and product, or calculate sum and product from two numbers.

Try:

help_outlineHow to Useexpand_more

Solving Diamond Problems

This calculator solves mathematics diamond problems. Enter any two values and the calculator finds the other two. The equations used in each step are displayed so you can understand the solution process.

Two Solving Modes:

Find Numbers: Given the sum and product, find the two numbers
Find Sum & Product: Given two numbers, calculate their sum and product

Working with Decimals:

The calculator supports decimal values. However, if you enter the product and sum where one or both contain decimals, the calculator may not find integer factor solutions. For best results with "Find Numbers" mode, use integer values for sum and product.

Understanding the Diamond:

In a diamond problem, the two numbers go on the left and right sides. Their product appears at the top, and their sum appears at the bottom. This layout helps visualize the relationship between multiplication and addition.

infoWhat is Diamond Problem Solver?expand_more

A diamond problem (also called a diamond puzzle) is a mathematical exercise where four numbers are arranged in a diamond shape. The two side numbers multiply to give the top number (product) and add to give the bottom number (sum). Given any two values, you can solve for the other two.

Diamond problems are commonly used in algebra education to develop number sense and prepare students for factoring quadratic expressions. When factoring x² + bx + c, you need two numbers that multiply to c and add to b—exactly what diamond problems train you to find.

The skill of finding two numbers from their sum and product is essential for factoring trinomials, solving quadratic equations, and understanding the relationship between roots and coefficients of polynomials.

functionsFormulaexpand_more

Finding Sum and Product (Easy):

Sum = A + B

Product = A × B

Simply add and multiply the two numbers

Finding Numbers from Sum & Product:

Given: Sum = S, Product = P

Solve: x² - Sx + P = 0

x = (S ± √(S² - 4P)) / 2

Uses the quadratic formula to find both numbers

Discriminant Check:

D = S² - 4P

If D < 0: No real solutions exist

If D = 0: Both numbers are equal (S/2)

If D > 0: Two distinct real solutions

lightbulbExamplesexpand_more

Example 1: Find Numbers (Sum=7, Product=12)

Solve: x² - 7x + 12 = 0

D = 49 - 48 = 1

x = (7 ± 1) / 2 = 4 or 3

Numbers: 3 and 4 (verify: 3+4=7, 3×4=12)

Example 2: Find Numbers (Sum=5, Product=6)

Solve: x² - 5x + 6 = 0

D = 25 - 24 = 1

x = (5 ± 1) / 2 = 3 or 2

Numbers: 2 and 3 (verify: 2+3=5, 2×3=6)

Example 3: Negative Numbers (Sum=-1, Product=-12)

Solve: x² + x - 12 = 0

D = 1 + 48 = 49

x = (-1 ± 7) / 2 = 3 or -4

Numbers: 3 and -4 (verify: 3+(-4)=-1, 3×(-4)=-12)

Example 4: Find Sum & Product (Numbers: 5 and 6)

Sum = 5 + 6 = 11

Product = 5 × 6 = 30

Diamond: Top=30, Bottom=11, Sides=5,6

quizFAQexpand_more

What if the calculator says 'no real solutions'?expand_more

Some sum and product combinations have no real number solutions. This occurs when the discriminant (S² - 4P) is negative. For example, sum=2 and product=5 gives D = 4-20 = -16, which is negative. No two real numbers can add to 2 and multiply to 5.

Can diamond problems have negative numbers?expand_more

Yes! If the product is negative, one number must be positive and one negative (opposite signs). If the product is positive but the sum is negative, both numbers are negative. For example, sum=-7 and product=12 gives -3 and -4.

How are diamond problems used in factoring quadratics?expand_more

To factor x² + bx + c, find two numbers that multiply to c (product) and add to b (sum). Those numbers complete the factorization: x² + 7x + 12 = (x + 3)(x + 4) because 3 and 4 multiply to 12 and add to 7.

Why don't decimals work well for finding numbers?expand_more

When you enter decimal values for sum and product, the resulting numbers are often irrational or complex decimals. The calculator can still solve these, but the solutions may not be the 'nice' integer answers typically expected in diamond problems.

What's the connection to Vieta's formulas?expand_more

Diamond problems illustrate Vieta's formulas: for a quadratic x² - Sx + P = 0 with roots r₁ and r₂, the sum of roots equals S (r₁ + r₂ = S) and the product of roots equals P (r₁ × r₂ = P). This is exactly the diamond relationship.