Question

How do use an algebraic method to find the exact roots of the equation `(x-2)/x+3x=-1`?

Answer 1

First observation is that `x!=0`

Explanation:

Now we can multiply everything by `x`:
`(cancelx(x-2))/cancelx+3x*x=-1*x`

Converting this into a standard quadratic equation:
`x-2+3x^2=-x->3x^2+2x-2=0`

Using: `x_(1,2)=(-B+-sqrt(B^2-4AC))/(2A)` we get:

`x_(1,2)=(-2+-sqrt(2^2-4*3*(-2)))/(2*3)`

`x_(1,2)=(-2+-sqrt(4+24))/6=(-2+-sqrt28)/6=(-2+-2sqrt7)/6`

`x_(1,2)=-1/3+-1/3sqrt7`