SOLUTION: Please, help me solve this problem : What is the sum of the squares of the roots of (4x^2 - 9)^4 - 10(4x^2 - 9)^2 + 9 = 0.

Let me start noticing that the polynomial is of degree 8 and has 8 roots that are real or/and complex numbers.

But do not be worried or scared.


First of all, there is NO NEED to solve the given equation and to find the roots explicitly.

Use the Vieta's theorem instead.


The Vietas's theorem says: 

    1) the sum of all the roots  

        = 

       is equal to the coefficient at  taken with the opposite sign and divided by the coefficient at .

       Now notice that the coefficient at  of the given polynomial is equal to 0 (to zero).

       Since I assumed about your level, I will not explain "WHY".  It is OBVIOUS.

       So,  = 0.


    2)  The sum of all the products of all pairs of the roots  is equal to the coefficient at   
        divided by the leading coefficient of the polynomial.

        And you need to calculate the coefficient at  and to find   in this way.


Thanks to Vieta, we know now everything that is needed to get the answer.


Indeed,  the sum of the squares of the roots  

          =  =  = .