Simplify: $\frac{x+y}{x-y}+ \frac{x-y}{x+y}-\frac{2(x²-y²)}{x²+y²}$

This is a class 10 Question From Simplification of Rational Expressions chapter of Unit Algebra (Mathematics). All the steps for the solutions are mentioned in the description below. If that's hard for you to navigate, you can always visit the facebook link given at the end of every posts. 

Solution:

Given,

= $\frac{x+y}{x-y}+ \frac{x-y}{x+y}-\frac{2(x²-y²)}{x²+y²}$

= $\frac{(x+y)(x+y)}{(x-y)(x+y)}+ \frac{(x-y)(x-y)}{(x+y)(x-y)}-\frac{2(x²-y²)}{x²+y²}$

= $\frac{(x+y)^2+(x-y)^2}{(x+y)(x-y)} -\frac{2(x²-y²)}{x²+y²}$

= $\frac{x^2+2xy+y^2+x^2-2xy+y^2}{x^2-y^2} -\frac{2(x²-y²)}{x²+y²}$

= $\frac{2(x^2+y^2)}{x^2-y^2} -\frac{2(x^2-y^2}{x^2+y^2}$

= $\frac{2(x^2+y^2)(x^2+y^2)}{(x^2-y^2)(x^2+y^2)} -\frac{2(x^2-y^2)(x^2-y^2)}{(x^2+y^2)(x^2-y^2)}$

= $\frac{2\{(x^2+y^2)^2 -(x^2-y^2)^2\}}{x^4-y^4}$

= $\frac{2(x^4+y^4+2x^2y^2-x^4-y^4+2x^2y^2)}{x^4-y^4}$

= $\frac{2(4x^2y^2}{x^4-y^4}$

= $\frac{8x^2y^2}{x^4-y^4}$


= Answer


Explanation to the above answer.


Step 1: Copy the same question given.

Step 2: If we want to take the LCM, we look for the common multiple terms or factors in the denominator. So, we analyse what would be the common denominator? It would probably be (a²-b²) and we know, (a²-b²)=(a+b)(a-b). So, we multiply the first two terms accordingly. Remember that we are taking the LCM of the first two terms only. 

Step 3: Denominators of the first two terms are the same, we add and subtract the terms in the numerator.

Step 4: We perform simple mathematical operations. Using {(a+b)² = a²+2ab+b²} and {(a-b)²= a²-2ab+b²}

Step 5: We write the result of the mathematical addition from step 4 taking 2 as a common factor. 

Step 6: Take the LCM of the remaining two terms of the expression. Here our LCM is x⁴-y⁴. So, we multiply the terms accordingly.

Step 7: We add the numerators of the two terms because their denominators are same.

Step 8: We expand the terms using {(a+b)² = a²+2ab+b²} and {(a-b)²= a²-2ab+b²}.

Step 9: We do simple addition and write our result. 

Step 10: We write our answer. 


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Question: Simplify: (x+y)/(x-y) + (x-y)/(x+y)  -2(x²-y²)/(x²+y²). | Simplification of Rational Expressions | SciPiPupil

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