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

1 Answer
Dec 8, 2015

Answer:

#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.