Solving $(A-B)^2$ involves expanding the expression using the binomial theorem. This is a fundamental algebraic technique that helps simplify and solve polynomial expressions. Let’s walk through the steps to solve $(A-B)^2$
Step-by-Step Solution
- Understand the Binomial Theorem
The binomial theorem provides a way to expand expressions that are raised to a power. For the expression $(A-B)^2$, we use the specific case of the binomial theorem for a power of 2.
- Apply the Binomial Expansion Formula
The binomial expansion formula for $(A-B)^2$ is:$(A-B)^2 = A^2 – 2AB + B^2$
- Break Down Each Term
Let’s break down each term in the expanded form:- $A^2$: This is the square of the first term, A.
- $-2AB$: This is twice the product of the first term (A) and the second term (B). The negative sign comes from the subtraction in the original expression.
- $B^2$: This is the square of the second term, B.
- Combine the Terms
After breaking down each term, we combine them to get the expanded form:$A^2 – 2AB + B^2$
Example
Let’s take an example to make it clearer. Suppose we have $(3-2)^2$
- Identify A and B: Here, A = 3 and B = 2.
- Apply the formula:
$(3-2)^2 = 3^2 – 2(3)(2) + 2^2$
- Calculate each term:
- $3^2 = 9$
- $2(3)(2) = 12$
- $2^2 = 4$
- Combine the terms:
$9 – 12 + 4 = 1$
So, $(3-2)^2 = 1$
Conclusion
By following these steps, you can expand and simplify any expression of the form $(A-B)^2$. This technique is useful in various areas of algebra and helps build a strong foundation for more complex mathematical concepts.