(x-y)-answer.jpeg "(x+y)(x-y) Answer")
Understanding (x + y)(x - y)
The expression (x + y)(x - y) is a common algebraic pattern known as the difference of squares. This pattern is important to recognize because it provides a shortcut for expanding and simplifying certain types of expressions.
The Pattern
The difference of squares pattern states that:
(x + y)(x - y) = x² - y²
In simpler terms, when you multiply two binomials where one has a plus sign and the other has a minus sign, the result is the square of the first term minus the square of the second term.
Why Does It Work?
You can understand why this pattern works by using the distributive property (also known as FOIL):
- First: x * x = x²
- Outer: x * -y = -xy
- Inner: y * x = xy
- Last: y * -y = -y²
Combining the terms, we get: x² - xy + xy - y². Notice that the middle terms, -xy and xy, cancel each other out, leaving us with x² - y².
Applications
The difference of squares pattern has many applications in algebra and beyond:
- Factoring: This pattern can be used to factor expressions that have the form of x² - y².
- Simplifying expressions: It simplifies expressions by eliminating the middle terms.
- Solving equations: It can be used to solve quadratic equations where one side is a difference of squares.
Example
Let's say we want to expand (2a + 3b)(2a - 3b). Using the difference of squares pattern:
(2a + 3b)(2a - 3b) = (2a)² - (3b)² = 4a² - 9b²
This is much faster than using the distributive property directly.
In Conclusion
Recognizing the difference of squares pattern is a valuable tool in algebra. It allows you to simplify expressions, factor polynomials, and solve equations efficiently. Mastering this pattern will make your algebraic journey smoother!