Prove the difference of squares formula.

The difference of squares formula states that (a+b)(a-b) = a^2 - b^2.

To prove the difference of squares formula, we can start by expanding the left-hand side of the equation:

(a+b)(a-b) = a(a-b) + b(a-b)

= a^2 - ab + ab - b^2

= a^2 - b^2

Therefore, we have shown that (a+b)(a-b) = a^2 - b^2, which is the difference of squares formula.

Another way to prove the difference of squares formula is to use algebraic manipulation. We can start with the right-hand side of the equation:

a^2 - b^2 = (a+b)(a-b)

Expanding the right-hand side gives:

a^2 - b^2 = a^2 - ab + ab - b^2

Simplifying this expression gives:

a^2 - b^2 = a^2 - b^2

Therefore, we have shown that the left-hand side of the equation is equal to the right-hand side, and thus the difference of squares formula is proven.

The difference of squares formula is a useful tool in algebra, as it allows us to simplify expressions involving squares of variables. For example, if we have the expression x^2 - 4, we can use the difference of squares formula to write it as (x+2)(x-2).