(x+y)^2+(x-y)^2 Formula

4 min read Jun 17, 2024

Understanding the (x+y)^2 + (x-y)^2 Formula

The formula (x+y)^2 + (x-y)^2 = 2(x^2 + y^2) is a useful algebraic identity that simplifies expressions involving squares of sums and differences. This article will explain the formula, its derivation, and some of its applications.

Derivation of the Formula

The formula can be derived using the basic algebraic identities for squaring binomials:

(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2

Let's substitute x for a and y for b:

(x+y)^2 = x^2 + 2xy + y^2
(x-y)^2 = x^2 - 2xy + y^2

Now, add these two equations together:

(x+y)^2 + (x-y)^2 = (x^2 + 2xy + y^2) + (x^2 - 2xy + y^2)

Simplifying the equation, we get:

(x+y)^2 + (x-y)^2 = 2x^2 + 2y^2

Finally, we arrive at the formula:

(x+y)^2 + (x-y)^2 = 2(x^2 + y^2)

Applications of the Formula

This formula has various applications in algebra and other areas of mathematics. Some of them include:

Simplifying expressions: The formula can be used to simplify expressions that involve squares of sums and differences. For example, simplifying (3+4)^2 + (3-4)^2 using the formula, we get 2(3^2 + 4^2) = 50.
Solving equations: The formula can be used to solve equations that involve squares of sums and differences.
Geometric applications: The formula can be used to derive relationships between sides and diagonals of squares and rectangles.

Examples

Here are some examples of how to use the formula:

Simplify (5+2)^2 + (5-2)^2

Using the formula, we get: 2(5^2 + 2^2) = 2(25 + 4) = 58

Solve the equation (x+3)^2 + (x-3)^2 = 34

Applying the formula, we get: 2(x^2 + 3^2) = 34. Simplifying this equation, we get x^2 + 9 = 17. Solving for x, we get x = ±√8.

Conclusion

The formula (x+y)^2 + (x-y)^2 = 2(x^2 + y^2) is a useful algebraic identity that simplifies expressions and solves equations. Understanding this formula will help you to manipulate algebraic expressions more effectively.