Question

Solve quadratic equation by Factorization :

$\frac{x - 1}{2x + 1} + \frac{2x + 1}{x - 1} = 2$

, x # - 1 / 2 , 1

Answer 1

Sorry by mistake ........

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Answer 2

Given equation : $\dfrac{x-1}{2x+1}+\dfrac{2x+1}{x-1}=2$

Let x - 1 = a and  2x + 1 = b

So, now equation is $\dfrac{a}{b}+\dfrac{b}{a}=2$

$\implies \bigg\{\dfrac{a}{b}\times\dfrac{a}{a}\bigg\}+\bigg\{\dfrac{b}{a}\times\dfrac{b}{b}\bigg\} = 2$

$\implies \dfrac{a^2}{ab} + \dfrac{b^2}{ab}=2$

$\implies \dfrac{a^2 +b^2}{ab}=2$

$\implies a^2 + b^2 = 2ab$

$\implies a^2 + b^2 - 2ab = 0$

$\mathsf{\texttt{ From the factorization, }a^2 + b^2 - 2ab = ( a - b )^2}$

$\therefore ( a - b )^2 = 0$

$\implies a - b = 0$

$\implies a = b$

Substituting the values of a and b : -

⇒ x - 1 = 2x + 1

⇒ - 1 - 1 = 2x - x

⇒ - 2 = x

Therefore the value of x is - 2.