Question

Which statement best reflects the solution(s) of the equation?

A. There is only one solution: x = 1. The solution x = 2 is an extraneous solution.


B. There is only one solution: x = 2. The solution x = 0 is an extraneous solution.


C. There is only one solution: x = 2. The solution x = 1 is an extraneous solution.


D. There are two solutions: x = 1 and x = 2.

Answer 1

C. There is only one solution:
`x = 2`. The solution
`x = 1` is an extraneous solution.

Explanation:

Given:

The equation is:

`(1)/(x-1)+(2)/(x)=(x)/(x-1)\\(x)/(x(x-1))+(2(x-1))/(x(x-1))=(x)/(x-1)\\(x+2x-2)/(x(x-1))=(x)/(x-1)\\(3x-2)/(x)=x\\ 3x-2=x^2\\x^2-3x+2=0`

Now, we solve this quadratic equation by factoring.

`x^2-3x+2=0\\(x-2)(x-1)=0\\x=2\textrm{ or }x=1`

Now,
`x=1` is an extraneous solution as at
`x` equal to 1, the equation is undefined.

Therefore,
`x=2` is the only solution. The solution
`x=1` is an extraneous solution.