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Simplify: $\dfrac{a+1}{a-1}+\dfrac{a^2-1}{a+1}$

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,

= $\dfrac{a+1}{a-1}+\dfrac{a^2-1}{a+1}$

= $\dfrac{(a+1)(a+1)}{(a-1)(a+1)} + \dfrac{(a^2-1)(a-1)}{(a+1)(a-1)}$

= $\dfrac{(a+1)^2}{a²-1} + \dfrac{(a^2-1)(a-1)}{a^2-1}$

= $\dfrac{(a+1)^2 + (a^2-1)(a-1)}{a^2-1}$

= $\dfrac{(a+1)(a+1) +(a+1)(a-1)(a-1)}{a^2-1}$

= $\dfrac{(a+1)\{(a+1) +(a-1)(a-1)\}}{(a+1)(a-1)}$

= $\dfrac{a+1+ a^2-2a+1}{a-1}$

= $\dfrac{a^2-a+2}{a-1}$

= Answer

Explanation to the above answer.


Step 1: Copy the same question given.

Step 2: To perform addition or subtraction, the denominators of two or more terms should be the same. In order to do this, we need to take the LCM of the denominators and convert them into like terms. So, we took LCM in this step.

Step 3: Now, we have the like denominators. 

Step 4: We add the given two terms. And write a single denominator.

Step 5: To shorten the length of the answer, we take the common factors out in the numerator. We used one formula: (a²-b²)=(a+b)(a-b)

Step 6: We take (a+1) common from the two terms and expand the denominator as well using the formula of a²-b².

Step 7: After dividing (a+1) in the numerator as well as in the denominator, we write the remaining expression. Also, when we multiply (a-1)(a-1), we get (a-1)² and (a-b)²= (a²-2ab+b²).

Step 8: We perform the basic mathematical operation (a-2a= -a) and get the remaining expression as answer.

Here is the Facebook link to the solution of this question in image. 

Here is the Website link to the guide of Simplification of Rational Expressions.


Question: Simplify: (a+1)/(a-1) + (a²-1)/(a+1). | Simplification of Rational Expressions | SciPiPupil

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