Question

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

Answer 1

Answer:

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

Step-by-step explanation:

Given:

The equation is:

$\frac{1}{x-1}+\frac{2}{x}=\frac{x}{x-1}\\\frac{x}{x(x-1)}+\frac{2(x-1)}{x(x-1)}=\frac{x}{x-1}\\\frac{x+2x-2}{x(x-1)}=\frac{x}{x-1}\\\frac{3x-2}{x}=x\\ 3x-2=x^2\\x^2-3x+2=0$

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.