For any quadratic equation of the form

,

the product of the two roots is .

Suppose the two roots are  and 

Then 

When he switched  and  in the
quadratic formula, he got the roots for

,

to be  and 

So their product  is ,

and that had to equal the number 

so 

So we have the system:



Clearing of fractions:



Solving each for the product 



So we have



cross-multiplying gives:



or



so, factoring,



So either  or 

Now in the first case, substitution of

of  for  in 



gives





---

In the second case, substitution of

of  for  in 



gives





So the product of the roots is either

 or 

Those are the answers!

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

If you want to demonstrate those answers in
specific cases, solve the quadratic equation:

 and you'll find the solutions

and  and  which have 
product 

Then switch  and  and you'll
get the equation

 and you'll find that it has
solutions   and  which 
are 3 times the solutions  and .



Edwin