When you encounter the expression $(A – B)^2$, you’re dealing with a binomial squared. This is a fundamental concept in algebra that helps simplify expressions and solve equations. Let’s break it down step by step.
Expanding $(A – B)^2$
The expression $(A – B)^2$ means $(A – B)$ multiplied by itself. In other words:
$(A – B)^2 = (A – B) times (A – B)$
To expand this, we use the distributive property (also known as the FOIL method for binomials):
$(A – B)(A – B) = A times A + A times (-B) + (-B) times A + (-B) times (-B)$
Now, let’s simplify each term:
- $A times A = A^2$
- $A times (-B) = -AB$
- $(-B) times A = -AB$
- $(-B) times (-B) = B^2$
Putting it all together, we get:
$A^2 – AB – AB + B^2$
Combine the like terms $-AB$ and $-AB$:
$A^2 – 2AB + B^2$
So, the expanded form of $(A – B)^2$ is:
$A^2 – 2AB + B^2$
Why Is This Useful?
Understanding how to expand $(A – B)^2$ is not just about memorizing a formula. It has practical applications in solving quadratic equations, simplifying expressions, and even in calculus. For example, if you need to solve the equation $(x – 3)^2 = 25$, knowing how to expand and simplify can make the process much easier.
Example Problem
Let’s say we have the expression $(5 – 2)^2$. Using our formula:
- $A = 5$
- $B = 2$
So, $(5 – 2)^2 = 5^2 – 2 times 5 times 2 + 2^2$
Simplify it step by step:
- $5^2 = 25$
- $2 times 5 times 2 = 20$
- $2^2 = 4$
Putting it all together:
$25 – 20 + 4 = 9$
So, $(5 – 2)^2 = 9$
Conclusion
The expression $(A – B)^2$ expands to $A^2 – 2AB + B^2$. This formula is a cornerstone in algebra, helping to simplify and solve various mathematical problems. Whether you’re working on quadratic equations or simplifying complex expressions, mastering this expansion is crucial for your mathematical toolkit.