# How do you solve y + 2/y = 1/y - 5?

Jun 11, 2015

Multiply through by $y$, rearrange into standard quadratic form and solve using the quadratic formula to give:

$y = \frac{- 5 \pm \sqrt{21}}{2}$

#### Explanation:

Given:

$y + \frac{2}{y} = \frac{1}{y} - 5$

First multiply both sides by $y$ to get:

${y}^{2} + 2 = 1 - 5 y$

(Note that in general multiplying both sides of an equation by $y$ could introduce a spurious solution of $y = 0$, but that won't happen in this case)

Add $5 y - 1$ to both sides to get:

${y}^{2} + 5 y + 1 = 0$

By the rational roots theorem we can immediately see that the only possible rational roots would be $y = \pm 1$, but neither of those works.

Let's use the quadratic formula.

Our quadratic equation is of the form $a {y}^{2} + b y + c = 0$, with $a = 1$, $b = 5$ and $c = 1$.

The discriminant is given by the formula:

$\Delta = {b}^{2} - 4 a c = {5}^{2} - \left(4 \times 1 \times 1\right) = 25 - 4 = 21$

Being positive but not a perfect square we can tell that our equations has two distinct irrational real roots.

The solutions are given by the formula:

$y = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{- 5 \pm \sqrt{21}}{2}$