Bonjour,
Une résolution classique:
[tex] 4x^2-(\sqrt{6} +4\sqrt{3} )x+\sqrt{18} =0\\
\Delta=(\sqrt{6} +4\sqrt{3} )^2-16*3\sqrt{2} \\
=54-24\sqrt{2} \\
[/tex]
[tex] \sqrt{54-24\sqrt{2}} =a+b\sqrt{c} \\
\Rightarrow\ 54-24\sqrt{2} =a^2+c*b^2+2ab\sqrt{c} \\
\Rightarrow\ c=2,\ a^2+2*b^2=54,\ ab=-12\\
\Rightarrow\ a^2+2(\dfrac{-12}{a})^2=54\\
\Rightarrow\ a^4-54a^2+288=0\\
\delta=54^2-4*288=1764=42^2\\
\Rightarrow\ a^2=\dfrac{54\pm\ 42}{2}\\
\Rightarrow\ a^2=48\ ou\ a^2=6\\
si\ a=\pm\sqrt{6}\ alors\ b=\mp\ 2\sqrt{6}\\
En\ r\'esum\'e\
si\ a=\pm\ 4\sqrt{3}\ alors\ b=\mp\ \sqrt{3}\\
[/tex]
[tex] \sqrt{54-24\sqrt{2}}=\sqrt{6}-4\sqrt{3}\\
\ ou\ \\
\sqrt{54-24\sqrt{2}}=-\sqrt{6}+4\sqrt{3}\\
x=\dfrac{\sqrt{6}+4\sqrt{3}+\sqrt{6}-4\sqrt{3}}{8}=\dfrac{\sqrt{6}}{4}\\
ou\ \\
x=\dfrac{\sqrt{6}+4\sqrt{3}-\sqrt{6}+4-4\sqrt{3}}{8}=\sqrt{3}\\
[/tex]
