Write -x^2 + 2xy - y^2 as the opposite of the square of a binomial.
AAAaannndd the bot gets it wrong yet again!
what a load of rubbish!
-x^2 + 2xy - y^2 = -(x^2- - 2xy + y^2) = -(x-y)^2
We can start by factoring the quadratic expression into two binomials:
-x^2 + 2xy - y^2 = (-x+y)(x-y)
To write this expression as the opposite of the square of a binomial, we just need to take the square root of each factor, put a minus sign in front, and multiply them together:
-(√(-x+y))^2 * (√(x-y))^2
Simplifying this expression, we have:
-(i√(x-y))^2 * (√(x-y))^2
-(i^2)*(x-y)
-(x-y)
Therefore, the answer is - (x-y).
To write -x^2 + 2xy - y^2 as the opposite of the square of a binomial, we need to complete the square.
Let's group the terms in pairs to see if we can identify a perfect square.
Grouping the terms, we have:
(-x^2 + 2xy) - y^2
Now, let's factor out the common factor from the first pair of terms:
x(-x + 2y) - y^2
To complete the square, we need to add and subtract the square of half the coefficient of the x term:
x(-x + 2y - (1/2)(-1)(-x)) - y^2
Simplifying,
x(-x + 2y + (1/2)x) - y^2
Now, let's factor out the common factor:
x( (1/2)x -x + 2y ) - y^2
Combining like terms inside the parenthesis,
x( (1/2)x - 2y) - y^2
Now, we can see that the expression can be written as the opposite of the square of a binomial:
-(x - (1/2)x + 2y)^2
Simplifying further,
-(x/2 + 2y)^2
Therefore, -x^2 + 2xy - y^2 can be written as the opposite of the square of the binomial (x/2 + 2y).
To write the expression -x^2 + 2xy - y^2 as the opposite of the square of a binomial, we need to identify a binomial that, when squared, will yield the expression in question. Let's go through the process step by step:
1. Start with the expression -x^2 + 2xy - y^2.
2. In order to make this expression the opposite of a square, we need to factor out a negative sign from each term:
-(-x^2 + 2xy - y^2)
This gives us x^2 - 2xy + y^2.
3. Now we can write this as the square of a binomial. To do this, we need to identify two terms that, when squared and combined, will give us x^2 - 2xy + y^2.
By comparing the expression to the perfect square binomial pattern (a - b)^2 = a^2 - 2ab + b^2, we can determine that our binomial will consist of:
a = x
b = y
4. Substituting a and b into the perfect square binomial pattern gives us:
(x - y)^2
5. Finally, to write the given expression as the opposite of the square of a binomial, we add the negative sign back:
-(-x^2 + 2xy - y^2) = -(x - y)^2
Therefore, the expression -x^2 + 2xy - y^2 can be rewritten as the opposite of the square of the binomial (x - y)^2.