what multiplies to but adds to

2 min read 26-12-2024

Finding two numbers that meet specific multiplication and addition criteria is a common problem in algebra and math puzzles. This guide will walk you through different methods to solve this, from simple guess-and-check to using quadratic equations. Understanding this concept is crucial for various mathematical applications.

Understanding the Problem

The core question is: What two numbers, when multiplied together, equal a target product (X), and when added together, equal a target sum (Y)? This problem often appears in factoring quadratic equations, solving word problems, and other mathematical contexts.

Methods for Solving

Let's explore several approaches to finding these numbers:

1. Guess and Check (For Smaller Numbers)

If the target product (X) and sum (Y) are relatively small, a simple guess-and-check approach can be effective. List the factor pairs of X and see which pair adds up to Y.

Example: Find two numbers that multiply to 6 and add to 5.

Factors of 6: (1, 6), (2, 3)
(2 + 3) = 5. Therefore, the numbers are 2 and 3.

This method becomes less efficient as the numbers get larger.

2. Using Algebra (For Larger Numbers)

For larger numbers, a more systematic algebraic approach is necessary. Let's represent the two unknown numbers as 'a' and 'b'. We can set up a system of two equations:

Equation 1: a * b = X (The product)
Equation 2: a + b = Y (The sum)

We can solve this system using substitution or elimination. Let's use substitution:

Solve Equation 2 for one variable: Let's solve for 'b': b = Y - a
Substitute: Substitute this expression for 'b' into Equation 1: a * (Y - a) = X
Expand and rearrange: aY - a² = X => a² - aY + X = 0
Solve the quadratic equation: This is a quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = -Y, and c = X. You can solve this using the quadratic formula:

a = [-b ± √(b² - 4ac)] / 2a

This will give you two possible values for 'a'. Substitute each value back into the equation b = Y - a to find the corresponding value for 'b'.

Example: Find two numbers that multiply to 12 and add to 7.

a * b = 12
a + b = 7
b = 7 - a
a * (7 - a) = 12
7a - a² = 12
a² - 7a + 12 = 0
Using the quadratic formula or factoring, we find a = 3 or a = 4.
If a = 3, then b = 7 - 3 = 4.
If a = 4, then b = 7 - 4 = 3.

Therefore, the two numbers are 3 and 4.

3. Factoring Quadratic Equations (Direct Application)

If the problem originates from a quadratic equation already in factored form (e.g., (x + a)(x + b) = 0), then 'a' and 'b' are your two numbers. The equation's constant term is the product (X), and the coefficient of the 'x' term is the sum (Y).

Real-World Applications

This concept has practical applications in various areas:

Physics: Solving kinematic equations involving velocity and time.
Engineering: Designing structures or circuits.
Finance: Calculating interest rates or investment returns.
Game Development: Creating puzzles or challenges.

Conclusion

Finding two numbers that multiply to X and add to Y can be approached in several ways. For smaller numbers, guess-and-check might suffice. However, for larger numbers, the algebraic approach using the quadratic formula provides a reliable and efficient solution. Understanding this principle is fundamental in numerous mathematical fields and real-world applications.