Understanding the Expression x^2 - y^2: A Step-by-Step Guide

Understanding the value of expressions in algebra is a fundamental skill that is often tested in various mathematical contexts. In this article, we will explore the expression x^2 - y^2 and explain how to calculate its value when specific values are assigned to x and y. Specifically, we will demonstrate the case where x 15 and y 5.

What is x^2 - y^2?

The expression x^2 - y^2 is a classical algebraic identity that can be simplified using the difference of squares formula. This identity is often used in various mathematical problems and is particularly useful in simplifying and solving equations.

The difference of squares formula states that:

x^2 - y^2 (x y)(x - y)

This identity allows us to break down the expression into simpler parts, making it easier to compute or manipulate.

Calculating x^2 - y^2 with Given Values

Given the values x 15 and y 5, we can now substitute these values into the expression to find the value of x^2 - y^2.

Step-by-Step Calculation

Substitute the values of x and y into the expression: x^2 - y^2 15^2 - 5^2 Calculate the squares: 15^2 225 5^2 25 Subtract the squares: 225 - 25 200 Therefore, the value of x^2 - y^2 when x 15 and y 5 is 200.

Verification Using an Algebraic Identity

To verify our result, we can use the algebraic identity (x y)(x - y) x^2 - y^2. Let's substitute the given values into this identity:

Calculate (x y) and (x - y): x y 15 5 20 x - y 15 - 5 10 Multiply these results: (20)(10) 200 This confirms that the value of x^2 - y^2 when x 15 and y 5 is indeed 200.

Understanding and Applying x^2 - y^2 in Different Contexts

The expression x^2 - y^2 is not only an interesting algebraic identity but also has various real-world applications. Here are some contexts where this identity is commonly used:

Geometry

Calculating distances in coordinate geometry.

Deriving the area of a trapezoid or parallelogram.

Understanding the Pythagorean theorem.

Physics

Calculating kinetic energy differences.

Understanding motion in curved paths.

Computer Science

Implementing algorithms for optimizing polynomial expressions.

Understanding data structures and algorithms involving quadratic equations.

Conclusion

The expression x^2 - y^2 is a powerful algebraic tool with wide-ranging applications. By understanding and applying the difference of squares identity, you can solve a variety of mathematical problems efficiently. Whether you are dealing with geometric calculations, physical phenomena, or computing algorithms, the knowledge of x^2 - y^2 will prove to be invaluable.