Giải phương trình: [tex]\frac{2+\sqrt{x}}{\sqrt{2}+\sqrt{2+\sqrt{x}}}+\frac{2-\sqrt{x}}{\sqrt{2}-\sqrt{2-\sqrt{x}}}=\sqrt{2}[/tex].
đặt [tex]\sqrt{2+\sqrt{x}}=a\\\\ \sqrt{2-\sqrt{x}}=b\\\\ => a^2+b^2=4\\\\ ab=\sqrt{4-x}\\\\ a-b=\sqrt{4-\sqrt{4-x}}=w[/tex]
[tex]\frac{2+\sqrt{x}}{\sqrt{2}+\sqrt{2+\sqrt{x}}}+\frac{2-\sqrt{x}}{\sqrt{2}-\sqrt{2-\sqrt{x}}}=\sqrt{2}\\\\ <=> \frac{a^2}{\sqrt{2}+a}+\frac{b^2}{\sqrt{2}-b}=\sqrt{2}\\\\ <=> \frac{a^2.(\sqrt{2}-b)+b^2.(\sqrt{2}+a)}{(\sqrt{2}+a).(\sqrt{2}-b)}=\sqrt{2}\\\\ <=> \sqrt{2}a^2-a^2b+\sqrt{2}b^2+ab^2=\sqrt{2}.(2+\sqrt{2}a-\sqrt{2}b-ab)\\\\ <=> \sqrt{2}.(a^2+b^2)-ab.(a-b)=2\sqrt{2}+2.(a-b)-\sqrt{2}ab\\\\ <=> 4\sqrt{2}-\sqrt{4-x}.\sqrt{4-2\sqrt{4-x}}=2\sqrt{2}+2.\sqrt{4-2\sqrt{4-x}}-\sqrt{2}.\sqrt{4-x}\\\\ <=> 2\sqrt{2}-(4-w^2).w=2.w-\sqrt{2}.(4-w^2)\\\\ <=> w^3-4w-2w-\sqrt{2}w^2+4\sqrt{2}+2\sqrt{2}=0\\\\ <=> w^3-\sqrt{2}w^2-6w+6\sqrt{2}=0\\\\ <=> w.(w-\sqrt{3}).(w+\sqrt{3})-\sqrt{2}.(w-\sqrt{3}).(w+\sqrt{3})=0\\\\ <=> (w-\sqrt{3}).(w+\sqrt{3}).(w-\sqrt{2})=0\\\\ <=> x=1 hoặc x=2[/tex]
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
