Multiple choice

If two prime number are the roots of the equation $5X ^2 +-25 X + a^2-a = 0 $ then the sum of possible value of a is

  1. 1

  2. -1

  3. 11

  4. -11
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A Correct answer
Explanation

Interpreting the garbled middle term as -25X, the sum of the roots is 25/5 = 5. The only pair of prime roots with sum 5 is 2 and 3, whose product is 6. Thus a^2 - a = 30, giving a = 6 or -5, and their sum is 1.

AI explanation

Let the prime roots be p and q; since their sum is -(-25)/5 = 5, the roots must be 2 and 3. Substitute X = 2 into the equation to get 20 + a^2 - a = 0, or (a - 4)(a + 3) = 0, yielding integer values of 4 and -3. Substitute X = 3 to get 45 + a^2 - a = 0, which yields no integer solutions for a. The sum of the possible integer values of a is 4 + (-3) = 1.

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