Multiple choice

A quadratic equation with integral coefficient has two different prime numbers as its roots if the sum of the coefficients of the equation is prime, then the sum of the roots is

  1. $2$
  2. $5$
  3. $7$
  4. $11$
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B Correct answer
Explanation

Let the roots be p and q (primes). The quadratic is (x-p)(x-q) = x^2 - (p+q)x + pq = 0. The sum of coefficients is 1 - (p+q) + pq = prime. This factors as (p-1)(q-1) = prime. Since p and q are primes, the only way for (p-1)(q-1) to be prime is if one factor is 1. Thus, p-1 = 1, so p = 2. Then (2-1)(q-1) = q-1 = prime. If q=3, q-1=2 (prime). Roots are 2 and 3. Sum = 5.

AI explanation

Let the prime roots be p and q, making the equation x squared minus the sum of p and q times x plus the product of p and q equal to zero. The sum of the coefficients is 1 minus the sum of p and q plus the product of p and q, which factors to 1 plus the quantity (p minus 1) times (q minus 1). For this to be prime, one of the roots must be 2, so the sum becomes 1 plus the quantity (q minus 1), which simplifies to q; the only way q is prime is if the roots are 2 and 3, making their sum 5. The result is 5.

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