Question

Solve the equation $x^4-10x^3+26x^2-10x+1=0$.

Answer 1

$x^4-10x^3+26x^2-10x+1=0$.

$\Rightarrow x^2-10x+26-\frac{10}{x}+\frac{1}{x^2}=0$, divide both sides by $x^2$

$\Rightarrow (x^2+\frac{1}{x^2})-10(x+\frac{1}{x})+26=0$

$\Rightarrow {(x+\frac{1}{x})}^2-2-10(x+\frac{1}{x})+26=0$

$\Rightarrow {(x+\frac{1}{x})}^2-10(x+\frac{1}{x})+24=0$

$\Rightarrow y^2-10y+24=0$, where $y=x+\frac{1}{x}$

$\Rightarrow (y-4)(y-6)=0$

$\Rightarrow y=4$ or $y=6$

$\Rightarrow x+\frac{1}{x}=4$ or $x+\frac{1}{x}=6$

$\Rightarrow x^2-4x+1=0$ or $x^2-6x+1=0$

$\Rightarrow x=2\pm \sqrt{3}$ or $x=3\pm \sqrt{8}$