Question

Fie multimea M={$\sqrt[2013]{ x + \sqrt{x^{2}-1 } } + \sqrt[2013]{x- \sqrt{ x^{2} -1} }=2$}
Daca S este suma tuturor elementelor din M luate la puterea a doua , atunci:
a)S=1; b)S=4; c) S=9 ; d)S=13 ; e)S=20 ; f)S=2

Answer 1

[tex](x+\sqrt{x^2-1})(x-\sqrt{x^2-1)} = 1 \\\;\\ \it x-\sqrt{x^2-1} =\dfrac{1}{x+\sqrt{x^2-1}}[/tex]

$\it Notam\ \sqrt[2013]{x+\sqrt{x^2-1}} = t$

Ecuatia devine:

$\it t+\dfrac{1}{t} =2 \Leftrightarrow t^2-2t+1=0\Leftrightarrow (t-1)^2=0 \Leftrightarrow t=1$

Deci:

$\it \sqrt[2013]{x+\sqrt{x^2-1}} =1 \Longrightarrow x = 1$