Question

If $\alpha and\beta$ are the roos of $x^2-px+q=0$ then ${\alpha }^4+{\beta }^4$ equal to

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

Answer 1

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

Given$x^2-px+q=0$

$\alpha +\beta =p,\alpha \beta =q$

${\alpha }^4+{\beta }^4={({\alpha }^2+{\beta }^2)}^2-2{\alpha }^2{\beta }^2$

${\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta =p^2-2q$

${\alpha }^4+{\beta }^4={(p^2-2q)}^2-2q^2$

$=p^4+4q^2-4p^{2}q$