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This problem trains for: SAT-I, AMC-8, GMAT.

The sum of the squares of the roots of the following equation:

is an integer divisible by:

Since:

the equation can be written:

Notice that we can factor a 3 out on the right hand side:

We can divide the equation by the factor 3-x:

keeping in mind that we have thus eliminated the choice x=3 since we cannot divide by zero. In this way, we may lose a root. Indeed, x=3 is a root of the equation (verify by plugging it in the original equation).

The other root(s) are given by:

The sum of the squares of the two roots is:

a number divisible by: 1, 3, 5, 9, 45 of which only 5 is among the answer choices.

The method we have outlined is designed to make the problem solvable mentally, without pen or calculator. We have written the steps down, but they are easy enough to be performed mentally. This is the fastest way of solving!

Otherwise, it can also be solved: