Answer:
[tex] \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2( {x}^{2} - {y}^{2}) }{ {x}^{2} - {y}^{2} } = \boxed{ \displaystyle \frac{4y ^2}{(x - y)(x + y)} }[/tex]
Step-by-step explanation:
we want to simplify the following
[tex] \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2( {x}^{2} - {y}^{2}) }{ {x}^{2} - {y}^{2} }[/tex]
notice that we can reduce the fraction thus do so:
[tex] \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2 \cancel{( {x}^{2} - {y}^{2}) }}{ \cancel{{x}^{2} - {y}^{2} }}[/tex]
[tex] \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - 2 [/tex]
in order to simplify the addition of the algebraic fraction the first step is to figure out the LCM of the denominator and that is (x-y)(x+y) now divide the LCM by the denominator of very fraction and multiply the result by the numerator which yields:
[tex] \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - 2 \\ \\ \displaystyle \frac{(x + y)^2 + (x - y)^2 - 2(x + y)(x - y)}{(x - y)(x + y)} [/tex]
factor using (a-b)²=a²+b²-2ab
[tex] \rm \displaystyle \frac{(x + y-(x - y) )^2}{(x - y)(x + y)} [/tex]
remove parentheses
[tex] \rm \displaystyle \frac{(x + y-x + y) )^2}{(x - y)(x + y)} [/tex]
simplify:
[tex] \rm \displaystyle \frac{4y ^2}{(x - y)(x + y)} [/tex]
9514 1404 393
Answer:
4y²/(x² -y²)
Step-by-step explanation:
The expression simplifies as follows:
[tex]\dfrac{x+y}{x-y}+\dfrac{x-y}{x+y}-\dfrac{2(x^2-y^2)}{x^2-y^2}\\\\=\dfrac{(x+y)(x+y)+(x-y)(x-y)-2(x^2-y^2)}{(x-y)(x+y)}\\\\=\dfrac{(x+y)^2+(x-y)^2-2(x^2-y^2)}{x^2-y^2}\\\\=\dfrac{(x^2+2xy+y^2)+(x^2-2xy+y^2)-2(x^2-y^2)}{x^2-y^2}\\\\=\dfrac{2(x^2+y^2-(x^2-y^2))}{x^2-y^2}=\boxed{\dfrac{4y^2}{x^2-y^2}}[/tex]
