SOLUTION: If p and q are the roots of the equation {{{2x^2-x-4=0}}} Find the equation whose roots are {{{p-(q/p)}}} and {{{q-(p/q)}}}

This assignment is for those who firmly knows the Vieta's theorem and Vieta's formulas.

    For the second degree polynomial (quadratic)  P(x)=   roots  and   of the equation  P(x)=0 satisfy

     +  =    and    = .     (1)     (see Wikipedia, this article).  


Concretely, for the given equation   = 0  its roots p and q satisfy 
     p + q =  and p*q = -2.     (2)

We can state it based in Vieta's theorem even without making explicit calculations of the roots.


Now, they want we construct the quadratic equation/polynomial with the roots  and .


1.  Then the coefficient at x of this polynomial, based on the Vieta's theorem,  must be the opposite number to

         +  =  -  =  - .       (3)

        From (2), you have  =  =  =  = .

        Therefore, we can continue and complete (3) in this way

         +  =  -  =  =  =  = .

        So, the opposite number to it,  is the coefficient at x in the polynomial under the question.


2.  Next step we should calculate the product . to determine the constant term of our polynomial.  The product is 

        . =  -  -  +  =  - .

        Now, replace pq by -2 and replace   by   =  =  =  =  = .

        You will get  . = ((-2) + 1) -  =  = .


Thus your equation under the question is   = 0.

Or, if you want to have it with integer coefficients, multiply everything by 16, and you will get

         = 0.

     The way I showed here, is THE ONLY WAY to solve the problems like this one.

     The problem is especially designed and intended to teach you this method.
     It is not my fantasy. It is the way you should know and the approach you must follow when solving such problems.

     It may happen you will have a desire to find the roots of the original equations and then manipulate with them.

     It is NOT THE WAY they want you follow in your solution.
     Saying "they", I mean those who assigned you this problem.

     They DEFINITELY want to know, whether you know the method I showed you, and
                                   whether you are able to use it.

HOW TO evaluate expressions involving  ,  ,    and