What is

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:

1. $A times A = A^2$
2. $A times (-B) = -AB$
3. $(-B) times A = -AB$
4. $(-B) times (-B) = B^2$

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:

1. $A = 5$
2. $B = 2$

So, $(5 – 2)^2 = 5^2 – 2 times 5 times 2 + 2^2$

Simplify it step by step:

1. $5^2 = 25$
2. $2 times 5 times 2 = 20$
3. $2^2 = 4$

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.

Citations

1. 1. Khan Academy – Expanding Binomials
2. 2. Math is Fun – Binomial Theorem
3. 3. Purplemath – Binomial Squares