Rewrite the expression

6j² - 6j - 12

                          Factor  6  out of all the terms

=   6(j² - j - 2)    (Notice here that if we distributed the 6, we would get back the previous expression.)

 

Now we want to force there to be a perfect square trinomial within the parenthesees. To do that, let's add and subtract half the coefficient of the middle term squared. That is, let's add and subtract  (1/2 * -1)²  which is  (-1/2)²  which is  1/4 . This will create a perfect square trinomial without changing the value of the expression.

 

=   6(j² - j + $\frac14$ - $\frac14$ - 2)

                                       Now there is a perfect square trinomial which is highlighted below:

=   6(j² - j + $\frac14$ - $\frac14$ - 2)

                                       Then  j² - j + 1/4  factors as  (j - 1/2)²

=   6( (j - $\frac12$)²  - $\frac14$ - 2)

                                       And  2 = 8/4

=   6( (j - $\frac12$)²  - $\frac14$ - $\frac84$)

                                       And -1/4 - 8/4  =  -9/4

=   6( (j - $\frac12$)²  - $\frac94$ )

                                       Distribute  6  to the terms in parenthesees

=   6( j - $\frac12$ )² -  6( $\frac94$ )

 

=   6( j - $\frac12$ )² -  $\frac{27}{2}$

 

Now it is in the form  c(j + p)² + q,  where   c = 6,  p =  -$\frac12$,  and  q = -$\frac{27}{2}$

 

So  q / p   =   ( -$\frac12$ ) / ( -$\frac{27}{2}$ )   =   ( -$\frac12$ ) * ( -$\frac{2}{27}$ )   =   $\frac{1}{27}$

Like Hecticar started, factor out 6 from the first two terms

6 (j^2 - j)     - 12      'complete the square' for j

6 (j-1/2)²   - 6/4 - 12

6 (j-1/2)² - 13 1/2