Question

How do you write $ \dfrac{4}{{(x + 2)}} + \dfrac{3}{{(x - 2)}} $ as a single fractional in its simplest form?

Answer 1

Hint: To solve this problem we should know about how to do operation in two fractional number.
Fractional number: it is used to represent part of the whole. It consists of a numerator and denominator. To solve this problem we should also be aware of algebraic identity to solve given problems.
  $ (x - a)(x + a) = {x^2} - {a^2} $

Complete step by step solution:
As given, here the fractional number is given.
To make given fractions into signal fractions first we add given two fractions. We get
  $ \dfrac{4}{{(x + 2)}} + \dfrac{3}{{(x - 2)}} = \dfrac{{4(x - 2) + 3(x + 2)}}{{(x + 2)(x - 2)}} $
By solving the above equation. We get,
  $ \Rightarrow \dfrac{{4(x - 2) + 3(x + 2)}}{{(x + 2)(x - 2)}} = \dfrac{{4x - 8 + 3x + 6}}{{(x + 2)(x - 2)}} $
  $ \Rightarrow \dfrac{{4x - 8 + 3x + 6}}{{(x + 2)(x - 2)}} = \dfrac{{7x - 2}}{{(x + 2)(x - 2)}} $
By applying algebraic identity. We get,
  $ \Rightarrow \dfrac{{7x - 2}}{{(x + 2)(x - 2)}} = \dfrac{{7x - 2}}{{{x^2} - {2^2}}} = \dfrac{{7x - 2}}{{{x^2} - 4}} $
So, $ \dfrac{4}{{(x + 2)}} + \dfrac{3}{{(x - 2)}} = \dfrac{{7x - 2}}{{{x^2} - 4}} $ it is a single fractional in its simple form.
So, the correct answer is “$\dfrac{{7x - 2}}{{{x^2} - 4}} $”.

Note: As we had solved it by many steps but in MCQ question. We can solve it by trying an error method by taking a value for $ x $ and putting the same value in the given option and selecting the one whose value is the same as the given question. This is used in engineering and different financing sectors to solve complicated problems and find a missing term.