Solving the Mathematical Puzzle: Two Numbers Add to 30 and Multiply to 144

This is the story of a seemingly straightforward problem that invites us to delve into the realms of algebra and quadratic equations. The challenge is to find two numbers that add up to 30 and, when multiplied, give a product of 144. Let's explore the solution step by step.

Introduction

The problem presented is a classic example of a system of equations: two numbers that, when added together, equal 30, and when multiplied, equal 144. Such problems serve as excellent exercises for strengthening one's algebraic problem-solving skills. They can be encountered in various forms of mathematics and real-life scenarios, making them a valuable addition to any problem collection.

Defining the Variables

Let’s denote the two numbers as a and b. According to the problem, these numbers satisfy the following conditions:

a b 30 ab 144

Our goal is to find the values of a and b that satisfy both equations.

Using Algebraic Manipulation

One way to approach this problem is by expressing one variable in terms of the other. From the first equation, b can be expressed as:

b 30 - a

Substituting this expression into the second equation, we get:

a(30 - a) 144

Expanding and rearranging the terms, we obtain a quadratic equation:

a^2 - 30a 144 0

Solving the Quadratic Equation

The next step is to solve this quadratic equation. This can be done using the quadratic formula or by factoring. Let's factor it:

a^2 - 24a - 6a 144 0

Grouping the terms, we get:

a(a - 24) - 6(a - 24) 0

Factoring out the common terms, we find:

(a - 24)(a - 6) 0

This equation has two solutions:

a - 24 0 rarr; a 24 a - 6 0 rarr; a 6

If a 24, then b 30 - 24 6. If a 6, then b 30 - 6 24. In both cases, the two numbers are 6 and 24.

Conclusion

The difference between these two numbers is:

24 - 6 18

Thus, the solution to the problem is that the two numbers are 6 and 24, and their difference is 18.

References

For a deeper dive into solving quadratic equations and other algebraic techniques, consider exploring:

Brian E. Blank and Steven G. Krantz, Calculus: Single Variable, 6th Edition Jack Degras, The Art of Problem Solving: A Resource for Top Mathematics Students, Inspiring Teachers and Coaches, and Entranced Learners

Related Keywords and Content

Math Problem: A collection of various mathematical puzzles and problems designed to challenge and improve problem-solving skills. Solving Quadratic Equations: Techniques and methods for solving equations of the form ax^2 bx c 0. Number Puzzle: Logical and mathematical puzzles that involve finding hidden patterns or solving equations to determine unknown values.