Solving a Pair of Numbers: Sum, Product, and Difference

Mathematics often involves understanding the relationships between numbers, particularly when dealing with their sums, products, and differences. This article will explore a problem involving a pair of numbers whose sum and product are given, specifically, a pair of numbers such that their sum is 5 and their product is 4. We aim to find the difference between these two numbers.

Understanding the Given Information

The problem can be formulated as follows: If the sum of two numbers is 5 and their product is 4, we need to determine the difference between them.

Solution via Direct Calculation

A straightforward approach to solve this problem is by direct calculation. Let's denote the two numbers as (a) and (b). We know the following:

Sum: (a b 5)

Product: (a times b 4)

By inspection or through simple guessing, we can find that the numbers are 4 and 1, since:

4 1 5

4 × 1 4

The difference between these numbers is:

(4 - 1 3)

Solving the Equation Algebraically

To provide a more general solution, let's set up the equations algebraically and solve for the difference between the two numbers using a quadratic equation. We begin with the given information:

Sum: (a b 5)

Product: (a times b 4)

From the sum equation, we can express one variable in terms of the other:

(b 5 - a)

Substitute this expression into the product equation:

((a) times (5 - a) 4)

This simplifies to:

(5a - a^2 4)

By rearranging, we get a standard quadratic equation:

(a^2 - 5a 4 0)

We solve this quadratic equation using the factoring method:

(a^2 - 4a - a 4 0)

((a - 4)(a - 1) 0)

This gives us the solutions:

(a 4)

(a 1)

Hence, the two numbers are 4 and 1. When (a 4) and (b 1), the difference is:

(4 - 1 3)

Alternative Method: Using a Shortcut Formula

Another interesting way to find the difference between the two numbers is by using the formula involving square roots, given the conditions (a b 5) and (ab 4).

The formula for the difference between the numbers is:

(a - b sqrt{a^2 b^2 - 2ab})

Note that:

((a b)^2 a^2 2ab b^2)

Given that (a b 5), we can substitute and solve for (a^2 b^2):

(5^2 a^2 2ab b^2)

(25 a^2 2(4) b^2)

(25 a^2 8 b^2)

(a^2 b^2 17)

Now, substitute into the difference formula:

(a - b sqrt{a^2 b^2 - 2ab} sqrt{17 - 8} sqrt{9} 3)

Thus, the difference between the two numbers is 3.

Conclusion

In conclusion, whether we use direct calculation, algebraic methods, or a shortcut formula involving square roots, we consistently arrive at the same result: the difference between the two numbers whose sum is 5 and product is 4 is 3.