Question

How do you solve the rational equation `1/6-1/x=4/(3x^2)`?

Answer 1

`x=3+-sqrt17`

Explanation:

Find a common denominator.

The denominators are `6,x,` and `3x^2`.

From this, we know the least common denominator will be `6x^2`.

Now, multiply each fraction so that the denominators all equal `6x^2`.

`1/6(x^2/x^2)-1/x((6x)/(6x))=4/(3x^2)(2/2)`

`x^2/(6x^2)-(6x)/(6x^2)=8/(6x^2)`

Multiply everything by `6x^2`, which will clear out the denominators altogether.

`x^2-6x=8`

`x^2-6x-8=0`

Use the quadratic formula or complete the square to solve for `x`, since the equation is not easily factorable:

`x=(6+-sqrt(36+32))/2=(6+-2sqrt17)/2=3+-sqrt17`

Return to the original equation to ensure that neither value of `x` will cause a fractional denominator to be `0`. Neither of the answers here will cause such an issue, but if one of the answers for `x` had been `0`, it would have had to been thrown out.