y+2/y=1/y-5

*Please solve the rational equation.

Done the same way as the previous post.

To solve the rational equation y + 2/y = 1/y - 5, we need to get rid of the fractions by finding a common denominator.

First, let's multiply the entire equation by the common denominator, which in this case is y.

So, we have:

y(y + 2/y) = y(1/y - 5)

Simplifying both sides of the equation, we get:

y^2 + 2 = 1 - 5y

Next, let's move all the terms to one side of the equation:

y^2 + 5y + 2 - 1 = 0

Combining like terms, we get:

y^2 + 5y + 1 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Since factoring this equation is not straightforward, let's use the quadratic formula:

The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = 5, and c = 1. Substituting these values into the quadratic formula:

y = (-5 ± √(5^2 - 4(1)(1))) / (2(1))

Simplifying further:

y = (-5 ± √(25 - 4)) / 2

y = (-5 ± √21) / 2

Therefore, the solutions to the equation are:

y = (-5 + √21) / 2

and

y = (-5 - √21) / 2