Question

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

• A. $p^3 - (3p - 1) q + q^2 = 0$
• 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

Answer: (A)

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

Answer 2

Given equation $x^2 + px + q = 0$ has roots $\alpha$ and $\alpha^2$ .
$\Rightarrow \alpha+ \alpha^{2} = - p$ and $\alpha^{3} = q$
$\Rightarrow \alpha\left(\alpha+1\right)=-p$
$\Rightarrow \alpha^{3}\left[\alpha^{3}+1+3\alpha\left(\alpha+1\right)\right]= - p^{3}$
$\Rightarrow q\left(q+1-3p\right) = -p^{3}$
$\Rightarrow p^{3} - \left(3p - 1\right)q + q^{2} = 0$