Multiple choice

A quadratic equation x2 + vx + 15 = 0 has two roots p and q, where p is of the form m8 + 324, (here, m is a non negative integer). For how many values of v, the given set of quadratic equations have one of the roots as prime numbers?

  1. 0

  2. 1

  3. 2

  4. 3
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A Correct answer
Explanation

The roots p and q satisfy p + q = -v and pq = 15. Since p = m^8 + 324, and m is a non-negative integer, the smallest value for p is 324 (when m=0). Since p is a root of x^2 + vx + 15 = 0, p must be a factor of 15 (1, 3, 5, 15). None of these values are possible for p >= 324, so there are no such values for v.

AI explanation

Because the product of the roots p and q is 15, p must be a divisor of 15. The given form p = m^8 + 324 yields values of 325, 325, 381, 1644, and so on for m = 0, 1, 2, 3. Since none of these numbers are divisors of 15, p can never be a prime root, resulting in 0 possible values for v.

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