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.