# What is x if -8=1/(3x)+x?

Nov 1, 2015

You have two solutions:
$x = - 4 - \sqrt{\frac{47}{3}}$, and
$x = - 4 + \sqrt{\frac{47}{3}}$

#### Explanation:

First of all, note that $x$ can't be zero, otherwise $\frac{1}{3 x}$ would be a division by zero. So, provided $x \setminus \ne 0$, we can rewrite the equation as

$\frac{3 x}{3 x} - 8 = \frac{1}{3 x} + x \frac{3 x}{3 x}$

$\iff$

$\frac{- 24 x}{3 x} = \frac{1}{3 x} + \frac{3 {x}^{2}}{3 x}$

with the advantage that now all the terms have the same denominator, and we can sum the fractions:

$\frac{- 24 x}{3 x} = \frac{1 + 3 {x}^{2}}{3 x}$

Since we assumed $x \setminus \ne 0$, we can claim that the two fractions are equal if and only if the numerators are equal: so the equation is equivalent to

$- 24 x = 1 + 3 {x}^{2}$

$3 {x}^{2} + 24 x + 1 = 0$.

To solve this, we can use the classic formula

$\setminus \frac{- b \setminus \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

where $a$, $b$ and $c$ play the role of $a {x}^{2} + b x + c = 0$.

So, the solving formula becomes

$\setminus \frac{- 24 \setminus \pm \sqrt{{24}^{2} - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

$=$

$\setminus \frac{- 24 \setminus \pm \sqrt{576 - 12}}{6}$

$=$

$\setminus \frac{- 24 \setminus \pm \sqrt{564}}{6}$

Since $564 = 36 \cdot \frac{47}{3}$, we can simplfy it out the square root, obtaining

$\setminus \frac{- 24 \setminus \pm 6 \sqrt{\frac{47}{3}}}{6}$

and finally we can simplify the whole expression:

$\setminus \frac{- \cancel{6} \cdot 4 \setminus \pm \cancel{6} \sqrt{\frac{47}{3}}}{\cancel{6}}$

into

$- 4 \setminus \pm \sqrt{\frac{47}{3}}$