POLYNOMIAL FRACTIONS : (2x)/(x-1) + (x-5)/(x^2-1) = 1

Question

I need help, I've been looking for hours now.

Answer 1

Note that x² - 1 = (x-1)(x+1).

 

Find a common denominator on the left hand side: x² - 1. This requires we multiply 2x/(x-1) by (x+1).

 

We then get:

 

2x(x+1)/(x² - 1) + (x-5)/(x² - 1) = 1

 

Combine the like terms on the left hand side:  ( 2x(x+1) + (x-5) ) / (x² - 1)  =  1

 

Multiply both sides by (x² - 1) and simplify the expression on the left hand side. You should get:

 

2x² + 3x - 5 = x² - 1

 

Combine like terms:  x² + 3x - 4 = 0

 

This is a standard quadratic function. Solve by completing the square, quadratic formula or factoring. I suggest factoring:

 

x² + 3x - 4 = (x-1)(x+4) = 0

 

This gives x = 1 and x = -4.

 

I leave this last part to you. Although we computed two x values, are both of them actually solutions? Why or why not?