Let $\alpha,\beta$ be the roots of the equation $x^2-x+2=0$ with $\text{Im}(\alpha)>\text{Im}(\beta)$. Then $\alpha^6+\alpha^4+\beta^4-5\alpha^2$ is equal to
Step-by-Step Solution
Key Concept: Roots of $x^2-x+2=0$: use $\alpha+\beta=1$, $\alpha\beta=2$. Repeatedly use $\alpha^2=\alpha-2$ (from the equation $\alpha^2-\alpha+2=0$) to reduce higher powers. Substitute $\beta=1-\alpha$ and simplify systematically.
$\alpha^2=\alpha-2\Rightarrow\alpha^4=\alpha^2-4\alpha+4=(\alpha-2)-4\alpha+4=2-3\alpha$.
$\alpha^6=\alpha^2\cdot\alpha^4=(\alpha-2)(2-3\alpha)=2\alpha-3\alpha^2-4+6\alpha=8\alpha-3(\alpha-2)-4=5\alpha+2$.
$\beta^4=2-3\beta$ (same formula with $\beta$).
$\alpha^6+\alpha^4+\beta^4-5\alpha^2=(5\alpha+2)+(2-3\alpha)+(2-3\beta)-5(\alpha-2)=5\alpha+2+2-3\alpha+2-3\beta-5\alpha+10=16-3\alpha-3\beta=16-3(1)=13$.
Correct Answer: 13