Multiple choice

If p and q are distinct prime numbers and if the equation $x^2-px+q=0$ has positive integers as its roots then the roots of the equation are

  1. 1, -1

  2. 2, 3

  3. 1, 2

  4. 3, 1
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C Correct answer
Explanation

For the quadratic equation x^2 - px + q = 0, the sum of roots is p and the product is q. Since p and q are primes and roots are positive integers, let the roots be r1 and r2. r1 * r2 = q implies one root must be 1 and the other q (since q is prime). Then r1 + r2 = 1 + q = p. For both p and q to be prime, the only solution is q = 2, which makes p = 3. Thus the roots are 1 and 2.

AI explanation

By Vieta's formulas for the equation x^2 - px + q = 0, the sum of the roots equals p and the product of the roots equals q. Since q is a distinct prime number, the only way its positive integer factors can multiply to q is if the roots are 1 and q. This makes the sum of the roots 1 + q, which implies the prime p equals q + 1. The only distinct prime numbers separated by 1 are 2 and 3, meaning the prime q is 2 and p is 3, so the roots are 1 and 2, which matches 1, 2.

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