Question

How do you solve `\frac { 1} { x ^ { 2} } - \frac { 1} { x } = 6`?

Answer 1

`x=-1/2,1/3`

Explanation:

We need the fractions to have a common denominator. We can get there through creative use of the number 1:

`1/x^2-1/x=6`

`1/x^2(1)-1/x(1)=6(1)`

`1/x^2(1/1)-1/x(x/x)=6(x^2/x^2)`

`1/x^2-x/x^2=(6x^2)/x^2`

And now we can multiply through by `x^2` which will eliminate the denominators, giving:

`1-x=6x^2`

`6x^2+x-1=0`

I'll use the quadratic formula to solve.

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

and we have `a=6, b=1, c=-1`

`x = (-1 \pm sqrt(1^2-4(6)(-1))) / (2(6))`

`x = (-1 \pm sqrt25) / 12=(-1pm5)/12`

`:. x=(-1+5)/12=4/12=1/3`
`:. x=(-1-5)/12=-6/12=-1/2`

Here's the graph:

graph{6x^2+x-1[-1,1,-2,2]}