If $alpha$ and $eta$ are the roots of the equation $x^{2}-x+3$ $=0$, then what is the value of $left(alpha^{4}+eta^{4} ight) ?$

If the roots of quadratic equation ax2 + bx + c

= 0 be D and Ethen

$alpha+eta=-frac{b}{a} ; alpha eta=frac{c}{a}$

$herefore$ For equation $x^{2}-x+3=0$,

$alpha+eta=1 ; alpha eta=3$

$herefore alpha^{4}+eta^{4}=left(alpha^{2} ight)^{2}+left(eta^{2} ight)^{2}=left(alpha^{2}+eta^{2} ight)^{2}-2 alpha^{2} eta^{2}$

$=left{(a+eta)^{2}-2 alpha eta ight}^{2}-2 alpha^{2} eta^{2}$

$=(1-2 imes 3)^{2}-2 imes 9=(-5)^{2}-18=25-18=7$