Multiple choice

If $p, q$ are the roots of equation $x^2 + px + q = 0$, then the value of $p$ must be equal to

  1. $0,1$
  2. $1$
  3. $2$
  4. $1, -2$
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A Correct answer
Explanation

For the equation x^2 + px + q = 0, the sum of roots is -p and the product of roots is q. Since the roots are p and q, we have p + q = -p (so 2p + q = 0) and pq = q. From pq = q, either q = 0 or p = 1. If q = 0, then 2p = 0, so p = 0. If p = 1, then 2 + q = 0, so q = -2. The values for p are 0 and 1.

AI explanation

By Vieta's formulas for the equation x^2 + px + q = 0 with roots p and q, the sum of the roots gives p + q = -p, so q = -2p. The product of the roots gives pq = q. Solving pq = q yields q = 0 or p = 1. Substituting q = 0 into q = -2p gives p = 0, and substituting p = 1 into q = -2p gives q = -2, meaning p can be either 0 or 1. The required values of p are 0 and 1.

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