Theory of Equations 1 Question 24
25. If one root is square of the other root of the equation $x^{2}+p x+q=0$, then the relation between $p$ and $q$ is
(a) $p^{3}-q(3 p-1)+q^{2}=0$
(2004, 1M)
(b) $p^{3}-q(3 p+1)+q^{2}=0$
(c) $p^{3}+q(3 p-1)+q^{2}=0$
(d) $p^{3}+q(3 p+1)+q^{2}=0$
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Solution:
- Let the roots of $x^{2}+p x+q=0$ be $\alpha$ and $\alpha^{2}$.
$$ \begin{array}{cc} \Rightarrow & \alpha+\alpha^{2}=-p \quad \text { and } \quad \alpha^{3}=q \\ \Rightarrow & \alpha(\alpha+1)=-p \\ \Rightarrow & \alpha^{3}{\alpha^{3}+1+3 \alpha(\alpha+1) }=-p^{3} \quad \text { [cubing both sides] } \\ \Rightarrow & q(q+1-3 p)=-p^{3} \\ \Rightarrow & p^{3}-(3 p-1) q+q^{2}=0 \end{array} $$
