How to Find

When you need to find the square of a difference between two variables, say $(A-B)^2$, you can use algebraic identities to simplify the process. Let’s break it down step by step.

Algebraic Identity

The algebraic identity for the square of a difference is:

$(A-B)^2 = A^2 – 2AB + B^2$

This formula helps you expand and simplify the expression. Let’s see how it works with an example.

Step-by-Step Example

Suppose we have $A = 5$ and $B = 3$. To find $(A-B)^2$, follow these steps:

1. Calculate $A^2$:

   $A^2 = 5^2 = 25$

2. Calculate $B^2$:

   $B^2 = 3^2 = 9$

3. Calculate $2AB$:

   $2AB = 2 times 5 times 3 = 30$

4. Substitute into the formula:

   $(A-B)^2 = 25 – 30 + 9$

5. Simplify the expression:

   $(A-B)^2 = 25 – 30 + 9 = 4$

So, $(5-3)^2 = 4$. This shows how the formula works in a practical example.

Why This Formula Works

The formula $(A-B)^2 = A^2 – 2AB + B^2$ is derived from the distributive property of multiplication over addition and subtraction. Here’s a quick proof:

1. Start with the expression $(A-B)(A-B)$
2. Use the distributive property to expand it:

   $(A-B)(A-B) = A(A-B) – B(A-B)$

3. Distribute $A$ and $B$:

   $= A^2 – AB – AB + B^2$

4. Combine like terms:

   $= A^2 – 2AB + B^2$

This confirms that the identity holds true for any values of $A$ and $B$

Practical Applications

Understanding how to find $(A-B)^2$ is useful in various mathematical contexts, including solving quadratic equations, analyzing functions, and even in physics for calculating distances or energy differences. For example, in physics, the formula might be used to determine the difference in energy levels between two states.

Conclusion

Finding $(A-B)^2$ using the given algebraic identity simplifies the process and ensures accuracy. By breaking it down into smaller steps and using the identity $A^2 – 2AB + B^2$, you can easily compute the square of a difference between two numbers.

Next time you encounter $(A-B)^2$, remember this handy formula and the steps to apply it correctly.

Citations

1. 1. Khan Academy – Algebra
2. 2. Math is Fun – Algebraic Identities
3. 3. Purplemath – Algebra Lessons