Solving for Two Numbers Given Their Sum and Product

Mathematics is a powerful tool for solving real-world problems, often involving simple yet intriguing questions. One such question is determining two numbers when their sum and product are known. In this article, we will explore how to solve for these numbers using algebraic methods, particularly focusing on quadratic equations. Let's dive in!

Understanding the Problem

The question at hand is: If the sum of two numbers is 11 and their product is 24, what are the two numbers?

Step-by-Step Solution

Let's denote the two numbers as x and y. The problem provides us with two key pieces of information:

The sum of the numbers (x y 11) Their product (xy 24)

We can solve these equations step-by-step as follows:

Step 1: Express y in terms of x

From the sum equation, we can express y as:

y 11 - x

Step 2: Substitute into the product equation

Substitute y 11 - x into the product equation xy 24:

x(11 - x) 24

Step 3: Expand and rearrange the equation

Expanding the equation gives us:

11x - x^2 24

Rearrange this to form a standard quadratic equation:

x^2 - 11x 24 0

Step 4: Factor the quadratic equation

The quadratic equation can be factored as:

(x - 3)(x - 8) 0

Step 5: Solve for x

Setting each factor to zero gives us the solutions for x:

x - 3 0 Rightarrow x 3

x - 8 0 Rightarrow x 8

Step 6: Find the corresponding values of y

If x 3, then y 11 - 3 8.

If x 8, then y 11 - 8 3.

Thus, the two numbers are 3 and 8.

Exploring Similar Problems

Let's consider a few more examples to solidify our understanding:

Example 1: Sum is 10, Product is 35

Using the same method, let the numbers be N and M such that:

N M 10 N × M 35

From this, we can express M as:

M 10 - N

Substitute M into the product equation:

N(10 - N) 35

Expand and rearrange:

N^2 - 10N 35 0

This can be factored as:

(N - 7)(N - 5) 0

Therefore:

N - 7 0 Rightarrow N 7

N - 5 0 Rightarrow N 5

Thus, the numbers are 7 and 5.

Example 2: Sum is 12, Product is 35

Again, using the same method, let the numbers be x and y such that:

x y 12 xy 35

From this, we can express y as:

y 12 - x

Substitute y into the product equation:

x(12 - x) 35

Expand and rearrange:

x^2 - 12x 35 0

This can be factored as:

(x - 7)(x - 5) 0

Therefore:

x - 7 0 Rightarrow x 7

x - 5 0 Rightarrow x 5

Thus, the numbers are 7 and 5.

Conclusion

By following the steps outlined in this article, you can solve for two numbers when their sum and product are given. The key to solving such problems lies in translating the given information into algebraic equations and then using quadratic equations to find the solutions. Whether you are a student, a teacher, or just someone with a curious mind, understanding these fundamental concepts will undoubtedly enhance your problem-solving skills.