Question

If $\alpha ,\beta ,\gamma$ are the roots of the equation $x^3+p{x}^2+qx+r=0$, then ${\alpha }^2+{\beta }^2+{\gamma }^2=$

• A. $p^2+2q$
• B. $-p^2+2q$
• C. $-p^2-2q$
• D. $p^2-2q$

Answer 1

Answer: (D)

$p^2-2q$

As $\alpha ,\beta ,\gamma$ are roots of $x^3+p{x}^2+qx+r=0$
Gives
$S_1=\alpha +\beta +\gamma =-p{S}_2=\alpha \beta +\beta \gamma +\gamma \alpha =q$
Now,
${\alpha }^2+{\beta }^2+{\gamma }^2={(\alpha +\beta +\gamma )}^2-2(\alpha \beta +\beta \gamma +\gamma \alpha )=(-p{)}^2-2q=p^2-2q$