Question

If one root is square of the other root of the equation $x^2+px+q=0$, then the relation between p and q is

• A. $p^3-q(3p-1)+q^2=0$
• B. $p^3-q(3p+1)+q^2=0$
• C. $p^3+q(3p-1)+q^2=0$
• D. $p^3+q(3p+1)+q^2=0$

Answer 1

The correct option is A

$p^3-q(3p-1)+q^2=0$

Let the roots of $x^2+px+q=0$ be $\alpha$ and ${\alpha }^2.$
$\Rightarrow \alpha +{\alpha }^2=-pand{\alpha }^3=q\Rightarrow \alpha (\alpha +1)=-p\Rightarrow {\alpha }^3\{{\alpha }^3+1+3\alpha (\alpha +1\left)\right\}=-p^3$ [cubing both sides]
$\Rightarrow q(q+1-3p)=-p^3\Rightarrow p^3-(3p-1)q+q^2=0$