Introduction

In this article, we will explore a mathematical problem that involves finding two numbers where the sum is 4 and the product is -5. This problem combines algebraic methods and graphical methods to solve the equations. We'll delve into the equations, solutions, and even use graphing to visualize the result.

Understanding the Problem

We need to find two numbers such that: Their sum is 4. Their product is -5. Let's denote these numbers by x and y. Therefore, we have the following equations: x y 4 xy -5

Solving Algebraically

We can express y in terms of x from the first equation:

y 4 - x

Next, we substitute this expression for y into the second equation:

x(4 - x) -5

Simplifying and rearranging this equation, we get a quadratic equation:

4x - x^2 -5

x^2 - 4x - 5 0

We can factor this quadratic equation as follows:

(x - 5)(x 1) 0

Setting each factor to zero gives us the possible values for x: x - 5 0 → x 5 x 1 0 → x -1 Now, we can find y for each value of x: If x 5, then y 4 - 5 -1. If x -1, then y 4 - (-1) 5. Thus, the two numbers are 5 and -1. It's easy to verify that these numbers satisfy the original conditions: 5 (-1) 4 5 * (-1) -5

Using Algebraic Factoring

Another way to solve this problem is to use a more direct algebraic approach. We can start with the quadratic equation derived earlier:

x^2 - 4x - 5 0

We can factor it as follows:

(x - 5)(x 1) 0

Again, setting each factor to zero gives us the solutions x 5 and x -1. Substituting these back into the expression for y, we find: If x 5, then y 4 - 5 -1. If x -1, then y 4 - (-1) 5.

Graphical Interpretation

A graphical method can also help us visualize the solution. Consider the two functions: y1 4 - x from x y 4 y2 -5/x from xy -5 The intersection of these two graphs will give us an estimate of the x and y values that satisfy both equations. In this case, the intersection occurs at (5, -1) and (-1, 5). Here's a brief recap of the graphical approach: Graph the function y 4 - x. Graph the function y -5/x. Identify the points where these graphs intersect. By plotting these functions, we can see that the points of intersection are indeed (5, -1) and (-1, 5).

Conclusion

In conclusion, the two numbers that add up to 4 and multiply to -5 are 5 and -1. Both the algebraic and graphical methods confirm this solution. Understanding how to solve such problems using different methods can be beneficial for enhancing problem-solving skills in mathematics.