If $p$ and $q$ are roots of the equation $\displaystyle x^{2}+px+q= 0$, then

Since p and q are the roots of the equation x^2 + px + q = 0, we can apply Vieta's formulas. The sum of the roots is p + q = -p and the product of the roots is p * q = q. From the product equation pq = q, we get q = 0 or p = 1. If q = 0, the sum equation becomes p = -p, so p = 0, but this makes the equation x^2 = 0, which lacks two distinct roots p and q. Thus, p must equal 1; substituting p = 1 into the sum equation gives 1 + q = -1, meaning q = -2. Therefore, the values are p = 1, q = -2.

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