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)

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