The largest root of $6x^4-35x^3+62x^2-35x+6=0$ is:

$x=0$ is not a root, so divide by $x^2:6(x^2+1/x^2)-35(x+1/x)+62=0$. With $t=x+1/x$ we have $x^2+1/x^2=t^2-2$, giving $6t^2-35t+50=0$, so $t=10/3$ or $t=5/2$. From $x+1/x=10/3:3x^2-10x+3=0$, so $x=3$ or 1/3. From $x+1/x=5/2:2x^2-5x+2=0$, so $x=2$ or 1/2. The largest root is 3.