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

Dec 8, 2015

$x = 3 \pm \sqrt{17}$

#### Explanation:

Find a common denominator.

The denominators are $6 , x ,$ and $3 {x}^{2}$.

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

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

$\frac{1}{6} \left({x}^{2} / {x}^{2}\right) - \frac{1}{x} \left(\frac{6 x}{6 x}\right) = \frac{4}{3 {x}^{2}} \left(\frac{2}{2}\right)$

${x}^{2} / \left(6 {x}^{2}\right) - \frac{6 x}{6 {x}^{2}} = \frac{8}{6 {x}^{2}}$

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

${x}^{2} - 6 x = 8$

${x}^{2} - 6 x - 8 = 0$

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

$x = \frac{6 \pm \sqrt{36 + 32}}{2} = \frac{6 \pm 2 \sqrt{17}}{2} = 3 \pm \sqrt{17}$

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.