Question

The largest positive root of $x^4+2x^3-22x^2+2x+1=0$ is

• A. $3+\sqrt{2}$
• B. $2+\sqrt{3}$
• C. $1+\sqrt{7}$
• D. $1+\sqrt{8}$

Answer 1

Answer: (B)

$2+\sqrt{3}$

Given equation $x^4+2x^3-22x^2+2x+1=0$
$x^2+\frac{1}{x^2}+2[x+\frac{1}{x}]-22=0$
$\Rightarrow {[x+\frac{1}{x}]}^2+2[x+\frac{1}{x}]-24=0$
Now put $t=x+\frac{1}{x}$
$\Rightarrow t^2+2t-24=0$
$\Rightarrow (t-4)(t+6)=0$
$\Rightarrow t=4ort=-6$.
Taking $t=4$
$4=x+\frac{1}{x}$
$x^2-4x+1=0$
$\therefore x=2+\sqrt{3},2-\sqrt{3}$
Taking $t=-6$
$-6=x+\frac{1}{x}$
$x^2+6x+1=0$
$\therefore x=-3-2\sqrt{2},-3+2\sqrt{2}$
Largest positive root is $2+\sqrt{3}$.