Can someone please check this to see if I have reduced to simplest terms:
{(-x-y)X(x-y)}/{(x-y)^2 + (x+y)^2}
I get -1/2 (x-y)(x+y)divided by (x^2+y^2).
Is this right and if so, can it be simplified any more? Thanks so much.
To check if the expression has been reduced to its simplest form, we can simplify it further.
First, let's expand the numerator: (-x - y)(x - y) = -x^2 + xy + xy - y^2.
Next, let's expand the denominator using the formula a^2 + b^2 = (a + b)(a - b): (x - y)^2 + (x + y)^2 = (x^2 - 2xy + y^2) + (x^2 + 2xy + y^2) = 2x^2 + 2y^2.
So now we have:
(-x^2 + xy + xy - y^2)/(2x^2 + 2y^2).
To simplify this expression, we can factor out a common factor in the numerator:
-x^2 + 2xy - y^2)/(2x^2 + 2y^2).
Now, let's simplify both the numerator and denominator individually:
Numerator:
-x^2 + 2xy - y^2 = -(x^2 - 2xy + y^2) = -(x - y)^2.
Denominator: 2x^2 + 2y^2 = 2(x^2 + y^2).
Substituting the simplified numerator and denominator, we get:
-(x - y)^2 / 2(x^2 + y^2).
So the expression (-x - y)(x - y) / [(x - y)^2 + (x + y)^2] simplifies to -(x - y)^2 / 2(x^2 + y^2).
Therefore, your simplification is correct, and there is no further simplification possible.