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

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