How can you write -x^2 + 2xy - y^2 as the opposite of the square of a binomial?
We can write -x^2 + 2xy - y^2 as the opposite of the square of a binomial by factoring it as the product of two identical binomials and then recognizing the binomial being squared.
First, notice that we can rewrite the given expression as: -(x^2 - 2xy + y^2). Now, we can factor the expression inside the parentheses as a perfect square trinomial. A perfect square trinomial can always be factored as (a - b)^2 or (a + b)^2, where a and b are the square roots of the first and third terms, respectively.
In our case, the square root of x^2 is x, and the square root of y^2 is y. Since the middle term is negative (-2xy), we will use the (a - b)^2 form. So, we can factor (x^2 - 2xy + y^2) as (x - y)^2.
Now, substituting this back into the original expression, we have: - (x - y)^2. So, -x^2 + 2xy - y^2 can be written as the opposite of the square of a binomial, which is - (x - y)^2.