Question

If $\alpha$ and $\beta$ are roots of $x^2+px+q=0$, then value of ${\alpha }^4+{\beta }^4$ in terms of $p$ and $q$ is

• A. $p^4-2q^2$
• B. $p^4-p^{2}q$
• C. $p^2+4p^{2}q$
• D. $p^4+2q^2-4p^{2}q$

Answer 1

The correct option is D $p^4+2q^2-4p^{2}q$

We know that ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
Substituting the values of $(\alpha +\beta )$ and $\alpha \beta$ in above equation, we get
${\alpha }^2+{\beta }^2=(-p{)}^2-2q=p^2-2q$.
We also know that ${\alpha }^4+{\beta }^4=({\alpha }^2+{\beta }^2{)}^2-2{\alpha }^2{\beta }^2=({\alpha }^2+{\beta }^2{)}^2-2(\alpha \beta {)}^2$
$=(p^2-2q{)}^2-2(q{)}^2=p^4+4q^2-4p^{2}q-2q^2=p^4+2q^2-4p^{2}q$.