This has a "surprise" answer:



The solutions by the quadratic equation are:

 

 

 

So there are two solutions:

(-k + sqrt( k^2+144) )/12 and (-k - sqrt( k^2+144) )/12

Any number times its negative reciprocal must equal -1.

So we multiply these two roots and set the product 
equal to -1:



Multiply using "FOIL"









Since this comes out an identity, that means that
 can be any number whatsoever, and in every
case one root will be the negative reciprocal of
the other!

Edwin