Επίθεση στις άρρητες 12
Δημοσιεύτηκε: Δευ Ιούλ 22, 2013 12:09 pm
Να λύσετε την εξίσωση: ![\sqrt[3]{{x - 2}} + \sqrt[3]{{x - 9}} = \sqrt[3]{{3\left( {2x - 11} \right)}}](/forum/ext/geomar/texintegr/latexrender/pictures/7c8d588c43a18c6865c4b5b57c0dee05.png "\sqrt[3]{{x - 2}} + \sqrt[3]{{x - 9}} = \sqrt[3]{{3\left( {2x - 11} \right)}}")
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Λύση
Έστω
μια λύση της δοσμένης εξίσωσης. Τότε, 
και![\displaystyle{\sqrt[3]{x-2}+\sqrt[3]{x-9}=\sqrt[3]{3\left(2\,x-11\right)}\Rightarrow \sqrt[3]{x-2}+\sqrt[3]{x-9}+\left[-\sqrt[3]{3\left(2\,x-11\right)}\right]=0}](/forum/ext/geomar/texintegr/latexrender/pictures/d9fed0f8ad6b804717ab7c1dc9baf864.png "\displaystyle{\sqrt[3]{x-2}+\sqrt[3]{x-9}=\sqrt[3]{3\left(2\,x-11\right)}\Rightarrow \sqrt[3]{x-2}+\sqrt[3]{x-9}+\left[-\sqrt[3]{3\left(2\,x-11\right)}\right]=0}")
Συνεπώς, από την ταυτότητα του
, προκύπτει,![\displaystyle{ x-2+x-9-3\left(x-2+x-9\right)=-3\,\sqrt[3]{3}\,\sqrt[3]{x-2}\,\sqrt[3]{x-9}\,\sqrt[3]{(x-2)+(x-9)}}](/forum/ext/geomar/texintegr/latexrender/pictures/b593644aaeb49a3f7fcf2341972d0dc1.png "\displaystyle{ x-2+x-9-3\left(x-2+x-9\right)=-3\,\sqrt[3]{3}\,\sqrt[3]{x-2}\,\sqrt[3]{x-9}\,\sqrt[3]{(x-2)+(x-9)}}")
![\displaystyle{\Rightarrow 2\left[\left(x-2\right)+\left(x-9\right)\right]=3\,\sqrt[3]{3}\,\sqrt[3]{x-2}\,\sqrt[3]{x-9}\,\sqrt[3]{(x-2)+(x-9)}}](/forum/ext/geomar/texintegr/latexrender/pictures/4887f8264ddc07f1cf12f93e61a817dd.png "\displaystyle{\Rightarrow 2\left[\left(x-2\right)+\left(x-9\right)\right]=3\,\sqrt[3]{3}\,\sqrt[3]{x-2}\,\sqrt[3]{x-9}\,\sqrt[3]{(x-2)+(x-9)}}")
![\displaystyle{\Rightarrow 8\left[\left(x-2\right)^3+\left(x-9\right)^3+3\left(x-2\right)^2\left(x-9\right)+3\left(x-2\right)\left(x-9\right)^2\right]=81\,\left(x-2\right)\left(x-9\right)\left[\left(x-2\right)+\left(x-9\right)\right]}](/forum/ext/geomar/texintegr/latexrender/pictures/30dc2d28aa2f725739f4dc1ca074a8da.png "\displaystyle{\Rightarrow 8\left[\left(x-2\right)^3+\left(x-9\right)^3+3\left(x-2\right)^2\left(x-9\right)+3\left(x-2\right)\left(x-9\right)^2\right]=81\,\left(x-2\right)\left(x-9\right)\left[\left(x-2\right)+\left(x-9\right)\right]}")
![\displaystyle{\Rightarrow 8\left[\left(x-2\right)^3+\left(x-9\right)^3\right]-57\left(x-2\right)^2\left(x-9\right)-57\left(x-2\right)\left(x-9\right)^3=0}](/forum/ext/geomar/texintegr/latexrender/pictures/4fe865c3269609365e168186c7a16653.png "\displaystyle{\Rightarrow 8\left[\left(x-2\right)^3+\left(x-9\right)^3\right]-57\left(x-2\right)^2\left(x-9\right)-57\left(x-2\right)\left(x-9\right)^3=0}")
![\displaystyle{\Rightarrow 8\left[\left(x-2\right)+\left(x-9\right)\right]\left[\left(x-2\right)^2-\left(x-2\right)\left(x-9\right)+\left(x-9\right_^2\right]-57\left(x-2\right)\left(x-9\right)\left[\left(x-2\right)+\left(x-9\right)\right]=0}](/forum/ext/geomar/texintegr/latexrender/pictures/88e3893cf626f011bb4fe2c99ce35e92.png "\displaystyle{\Rightarrow 8\left[\left(x-2\right)+\left(x-9\right)\right]\left[\left(x-2\right)^2-\left(x-2\right)\left(x-9\right)+\left(x-9\right_^2\right]-57\left(x-2\right)\left(x-9\right)\left[\left(x-2\right)+\left(x-9\right)\right]=0}")
Δεκτή είναι μόνο η τιμή
που επαληθεύει και άρα είναι η μοναδική λύση της αρχικής εξίσωσης.edit:Προσθήκη περιορισμού