We know that x=0 doesn't satisfy the equation, therefore we can simply do this.
6x^4-35x^3+62x^2-35x+6=0 \\ =>x^2(6(x^2+1/x^2)-35(x+1/x)+62)=0

now take x+1/x=y therefore you get 

x^2(6(y^2-2)-35y+62)=0 \\ =>x^2(6y^2-35y+50)=0 \\ =>x^2(6y^2-15y-20y+50)=0 \\ =>x^2(2y-5)(3y-10)=0

now put back y=x+1/x and multiply each factor by x, you get

(2x^2-5x+2)(3x^2-10x+3)=0 \\ =>(x-2)(2x-1)(x-3)(3x-1)=0 \\ =>x=2,3,1/2,1/3.
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First divide the complete equation by x².
then u will get,6x²-35x+62-35/x+6/x²=0
  now put x+1/x=a----->eq.2
then squaring on both sides,
now sub. the values in eq.1
then by solving u will get two values of a.
than sub. the two values in eq.2 and u will get four values for 'x'.
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