Two Numbers That Add to and Multiply to Calculator

Find two numbers based on their sum and product.

About This Solver

This calculator solves a classic algebra problem often encountered in mathematics known as the "Sum and Product" puzzle or the "Diamond Problem." The goal is to find two unknown numbers, let's call them x and y, that satisfy two specific conditions simultaneously:

1. They must add up to a specific Sum ($x + y = S$).
2. They must multiply to a specific Product ($x \cdot y = P$).

How It Works (The Math)

While you can often solve simple versions of this problem by guessing and checking factors, this calculator uses the quadratic formula to find the exact answer instantly, even for decimals or large numbers.

Mathematically, if we know that $x + y = S$ and $xy = P$, these two numbers are actually the roots of the quadratic equation:

x² – Sx + P = 0

By substituting your Sum ($S$) and Product ($P$) into the Quadratic Formula, we can solve for $x$:

x = [ S ± √( S² – 4P ) ] / 2

Discriminant Analysis

The term inside the square root ($S^2 – 4P$) is called the discriminant. It tells us about the nature of the solution:

• Positive: Two distinct real numbers exist.
• Zero: One unique real number exists (both numbers are the same).
• Negative: The solution involves complex/imaginary numbers (no real solution).

Common Applications

1. Factoring Quadratics

This is the most common use case. When factoring a trinomial in the form $ax^2 + bx + c$, you often need to find two numbers that add up to $b$ and multiply to $ac$. This is a crucial step in the "AC Method" or "Splitting the Middle Term."

2. The Diamond Problem

In many algebra curriculums, students use a visual aid called a "Diamond Problem" where the product is at the top, the sum is at the bottom, and the student must fill in the left and right corners with the two numbers that satisfy the condition.

Example Calculation

Suppose you want to find two numbers that add to 12 and multiply to 35.

• Sum (S): 12
• Product (P): 35
• Logic: We look for factors of 35: (1, 35), (5, 7). Which pair adds to 12? 5 + 7 = 12.
• Result: The numbers are 5 and 7.

The discriminant (" + S + "² – 4*" + P + " = " + discriminant + ") is negative, meaning the answers are imaginary numbers.

x² – (" + S + ")x + (" + P + ") = 0