$ \alpha $ and $ \beta $ are the roots of the equation $ x^{2}+px+p^{3}=0 $ , $ p\neq0 $ . If the point $ (\alpha, \beta) $ lies on the curve $ x=y^{2} $, then the roots of the given equation are

The roots alpha and beta must satisfy the curve x = y^2, so alpha = beta^2. From the quadratic equation x^2 + p x + p^3 = 0, we know alpha + beta = -p and alpha beta = p^3. Substituting alpha = beta^2 yields beta^2 + beta = -p and beta^3 = p^3, so beta = p. Substituting beta = p into beta^2 + beta = -p gives p^2 + 2p = 0, yielding p = -2 (since p is not 0). The roots are beta = -2 and alpha = beta^2 = 4, making the roots 4 and -2.

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