Question

Which solution to the equation 1/x-1=x-2/2x^2-2 is extraneous?

Answer 1

An extraneous solution is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the original problem.

Given the equation:

`(1)/(x-1) = (x-2)/(2x^2-2) \\ \\ 2x^2-2=(x-1)(x-2)=x^2-3x+2 \\ \\ x^2+3x-4=0 \\ \\ (x-1)(x+4)=0 \\ \\ x-1=0 \ or \ x+4=0 \\ \\ x=1 \ or \ x=-4`

From the equation, it is obtained that x = 1 and x = -4 are the solutions of the equation.

Now, we substitute x = 1 into the equation as follows:

`(1)/(1-1) = (1-2)/(2(1)^2-2) \\ \\ (1)/(0) = (-1)/(2-2) \\ \\ (1)/(0) = (-1)/(0)`

As can be see, for x = 1, the equation is undefined.

Now, we substitute x = -4 into the equation as follows:

`(1)/(-4-1) = (-4-2)/(2(-4)^2-2) \\ \\ (1)/(-5) = (-6)/(2(16)-2) = (-6)/(32-2) = (-6)/(30) \\ \\ - (1)/(5) =- (1)/(5)`

It can be seen that x = -4 is a valid solution of the orignal equation.

Therefore, x = 1 is an extraneous solution to the equation

`</span><span>(1)/(x-1) = (x-2)/(2x^2-2)`