Find the number of the roots of the equation

[Math Tutor]

Find the number of the roots of the equation:
[tex]x^4+1=2(2x-1)^{\frac{1}{4}}[/tex]

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The problem is thought out by r2d2 - professor in mathematics

[dduclam]

Find the number of the roots of the equation:
[tex]x^4+1=2(2x-1)^{\frac{1}{4}}[/tex] (1)

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The problem is thought out by r2d2 - professor in mathematics

My solution: condition [tex]x> \frac1{2}[/tex]

(1)<=>[tex]x^4+1=2\sqrt[4]{2x-1}[/tex] (2)

Put [tex]y=\sqrt[4]{2x-1} >0 => y^4=2x-1[/tex] (3)

(2) <=> [tex]x^4=2y-1[/tex] (4)

From (3) and (4) => [tex]x^4-y^4=2(y-x) <=> (x-y)[(x+y)(x^2+y^2)+2]=0[/tex]

<=> [tex]x=y[/tex] (because [tex]x>\frac1{2};y>0[/tex])

Thus,we only need solve equation: [tex]x^4-2x+1=0[/tex] ,which is not so hard