Question

solve for x, x-1/2x+1 + 2x+1/x-1 =2 , where does not = -1/2,1

Answer 1

The difference of numerators and denominators of the terms on L.H.S. :-

2x + 1 - x + 1 = x + 2

Similarly,2x + 1 - x + 1 = x +2

For this, the difference should be equal or proportional and here as we see the difference is equal.

Thus by putting one of the difference equals to zero, we can find the value of the variable.

Hence, x + 2 = 0
=> x = -2

This is a quadratic equation. So, there will be two roots, where each is -2

These types methods are only for fast solving.

Hope this will help you and you will use it....

Answer 2

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

On cross-multiplication, we get

$\frac{(x - 1) * (x - 1)+ (2x + 1)(2x + 1)}{(x - 1)(2x + 1)} = 2$

$\frac{(x - 1)^2 + (2x + 1)^2}{(x - 1)(2x + 1)} = 2$

We know that (a - b)^2 = a^2 + b^2 - 2ab, (a + b)^2 = a^2 + b^2 + 2ab

$\frac{(x^2 + 1 - 2x) + (4x^2 + 1 + 4x)}{(x - 1)(2x + 1)} = 2$

$x^2 + 1 - 2x + 4x^2 + 1 + 4x = 2(x - 1)(2x + 1)$

5x^2 + 2x + 2 = 2(2x^2 + x - 2x - 1)

5x^2 + 2x + 2 = 4x^2 + 2x - 4x - 2

5x^2 + 2x + 2 = 4x^2 - 2x - 2

x^2 + 4x + 4 = 0

(x + 2)^2 = 0

x = -2.


Therefore the value of x = -2.


Hope this helps!