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

Given equation $x^2+px+q=0$ has roots $\alpha$ and ${\alpha }^2$.

$\Rightarrow Sum=\alpha +{\alpha }^2=-p$

and $Product={\alpha }^3=q$

$\Rightarrow \alpha (\alpha +1)=-p$

$\Rightarrow {\alpha }^3[{\alpha }^3+1+3\alpha (\alpha +1\left)\right]=-p^3$

$\Rightarrow q(q+1-3p)=-p^3$

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

Hence, option A is correct.