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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:

equation

is an integer divisible by:

Since:

equation

the equation can be written:

equation

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

equation

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

equation

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:

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equation

equation

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The sum of the squares of the two roots is:

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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:

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