Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2-x-1=0$, then what is the value of $({\alpha }^4+{\beta }^4)$?

• A. $7$
• B. $0$
• C. $2$
• D. None of the above

Answer 1

Answer: (A)

$7$

Given equation is $x^2-x-1=0$

Let $\alpha ,\beta$ are the roots of $x^2-x-1=0$
Therefore, $\alpha +\beta =1,\alpha \beta =-1$
${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
$=1+2=3$
${\alpha }^4+{\beta }^4=({\alpha }^2+{\beta }^2{)}^2-2{\alpha }^2{\beta }^2$
$=3^2-2\times 1=7$

Thus value of $({\alpha }^4+{\beta }^4)$ is $7$.