SOLUTION: The equation {{{x^2-3x-2=0}}} has roots {{{alpha}}} and {{{beta}}}, and the equation {{{x^2-6x+p=0}}} has roots {{{k/alpha}}} and {{{k/beta}}}. find the value of {{{k}}} and of {{{

We need to know two things about a quadratic equation of the
form 

1.   the sum of the two roots with the sign changed.

2.   the product of the two roots.

We use 1 and 2 on the first equation:

Since  has roots  and ,

Using 1,  
multiplying both sides by 


Using 2,  

Now we use 1 and 2 on the second equations:

Since  has roots  and . 

Using 1,  
multiplying both sides by 

getting the LCD of 



Combine numerators over the LCD:



Factor out  on top:



Now,

since we have above that ,
and since , we can replace
 by ,

and
since we have above that ,
we can replace  by -2,




Multiply both sides by 

Divide both sides by 


So the value of  is 

-----------------

Using 2 on the second equation, 
 
But since ,



Now since from above, we have ,

 becomes





So the value of  is 

Edwin