Question

How do you solve `(2x)/1 = 1/(x-1)`?

Answer 1

`x=1/2(1+-sqrt3)`

Explanation:

`(2x)/1=1/(x-1)`

Cross multiply

`2x(x-1)=1`

`2x^2-2x-1=0`

For a quadratic equation of the form: `ax^2+bx+c`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

In our case: `a=2, b=-2, c=-1`

`:. x=(+2+-sqrt(4+4*2*1))/(2*2)`

`=(+2+-sqrt(12))/(4)`

`=(2+-2sqrt(3))/(4)`

`= 1/2(1+-sqrt3)`

Answer 2

Multiply through by the denominator of the right-hand side and then solve the resulting quadratic equation.

`x=1.366` or `-0.366`

Explanation:

`(2x)/1` is just `2x`, so

`2x=1/(x-1)`

Multiply both sides by `(x-1)`

`2x(x-1)=1(cancel(x-1))/cancel(x-1)`

`2x^2-2x=1`

`2x^2-2x-1=0`

Now we can use the quadratic formula (or any other method) to solve this quadratic equation.

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`=(2+-sqrt(4+8))/(4)`

`x=1.366` or `-0.366`