Question

If $\alpha$ and $\beta$ are roots of equation $x^2-\left(6+3^{\log _3 2}-2^{\log _2 3}\right) x+\left(6+2^{\sqrt{\log _2 3}}-3^{\sqrt{\log _3 2}}\right)=0$ then find the value of $2(\alpha+\beta)-\alpha \beta$.

Answer 1

Given equation becomes $x^2-5 x+6=0$ and $\alpha=2, \beta=3$
$\therefore 2(\alpha+\beta)-\alpha \beta=2(2+3)-2 \cdot 3=10 \cdot 6=4$