Question 37297
Given a number x, the reciprocal of x is 1/x.

6 times the reciprocal of x, is 6/x
and the problem says that the sum of x and its reciprocal is equal to -5,then:

x+6/x=-5  Now we have to eliminate the denominator by multiplying the entire expression by x

x(x)+x(6/x)=x(-5)
x^2+6=-5x
x^2+5x+6=0 Now we have to solve the equation to solve the problem and find the solutions.

The solution is given using the general formula for quadratic equations:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

Solutions: x1=-2,x2=-3
We have to solutions and therefore two pair of number for which the stated condition hold.
Those numbers are:
-2 and -1/2 , -3 and -1/3

Proof:

-2+6(-1/2) = -2-3=-5 
-3+6(-1/3) = -3-2=-5

Then the solutions are true.
Any doubt?:)