For a given value of k, the product of the roots of x^2-3kx+2k^2-1=0 is 7. The roots may be characterized as:

1960 AMC 12 Problem 11

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11.

For a given value of

k,

the product of the roots of

x^2-3kx+2k^2-1=0

7.

The roots may be characterized as:

integral and positive

integral and negative

rational, but not integral

irrational

imaginary

Use the constant term to determine the possible values of

k

For either value of

k,

examine the quadratic formula or discriminant

By Vieta’s formulas,

2k^2-1=7,

k=\pm2.

The roots are

3\pm\sqrt2

when

k=2,

and

-3\pm\sqrt2

when

k=-2.

In either case both roots are irrational.

Thus, the correct answer is D.