x₀² = 8 - ( \(2\sqrt{2+\sqrt{3}}\)+ \(2\sqrt{6-3\sqrt{3}}\)) (1)

Ta có (  \(2\sqrt{2+\sqrt{3}}\)+ \(2\sqrt{6-3\sqrt{3}}\))² = 32

Do đó x₀² = 8 - \(\sqrt{32}\)(2)

PT <=> (x² - 8)² - 32 = 0 (3)

Thế (2) vào (3) thì đúng

Vậy x₀ là nghiệm của PT

Đặt \(x^2=t\left(t\ge0\right)\)

\(\Leftrightarrow t^2-16t+32=0\)

\(\Delta=\left(-16\right)^2-4.32=256-128=128>0\)

\(t_1=\frac{16-\sqrt{128}}{2}=8-4\sqrt{2};t_2=\frac{16+\sqrt{128}}{2}=8+4\sqrt{2}\)

Theo bài ra ta có : 

\(x_0=\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\)

\(=\sqrt{2+\sqrt{3}}-\sqrt{3\left(2-\sqrt{2+\sqrt{3}}\right)}\)

tịt lun, cái pt căn này chill quá 

 ๖²⁴ʱ๖ۣۜTɦủү❄吻༉ Mơn Bạn nha .

P/s : làm nháp thử mn sửa giúp nha ( thực ra em cũng chả hiểu cái gì cả T_T )

Ta có :

\(\left(x_0\right)^2=8-2\sqrt{2+\sqrt{3}}-2\sqrt{3\left(2-\sqrt{3}\right)}\)

\(\Rightarrow\left(\frac{8-\left(x_0\right)^2}{2}\right)^2=2+\sqrt{3}+3\left(2-\sqrt{3}\right)+2\sqrt{3\left(4-3\right)}=8\)

\(\Rightarrow64-16\left(x_0\right)^2+\left(x_0\right)^4=32\)

\(\Rightarrow\left(x_0\right)^4-16\left(x_0\right)^2+32=0\left(đpcm\right)\)

ta có \(8-2\sqrt{3}+2\sqrt{3}=\left(2\cdot\sqrt{2}\right)^2\)

\(\Leftrightarrow\left(2+\sqrt{3}\right)+\left(6-3\sqrt{3}\right)+2\sqrt{\left(2+\sqrt{3}\right)\left(6-3\sqrt{3}\right)}=\left(2\cdot\sqrt{2}\right)^2\)

\(\Leftrightarrow\left(\sqrt{2+\sqrt{3}}+\sqrt{6-3\sqrt{3}}\right)^2=\left(2\cdot\sqrt{2}\right)^2\)

\(\Leftrightarrow\sqrt{2+\sqrt{3}}+\sqrt{6-3\sqrt{3}}=2\sqrt{2}\)

\(\Leftrightarrow8-2\sqrt{2+\sqrt{3}}-2\sqrt{\left(2+\sqrt{2+\sqrt{3}}\right)\left(6-\sqrt{2+\sqrt{3}}\right)}=8-4\sqrt{2}\)

\(\Leftrightarrow\left(\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\right)^2=8-4\sqrt{2}\)

\(\Leftrightarrow x_0^2=8-4\sqrt{2}\)

\(\Leftrightarrow x_0^2-\left(8-4\sqrt{2}\right)=0\)

\(\Leftrightarrow\left[x_0^2-\left(8-4\sqrt{2}\right)\right]\left[x_0^2-\left(8+4\sqrt{2}\right)\right]=0\)

\(\Leftrightarrow x_0^4-16x_0^2+32=0\)

Ta có:

\(x_0^2=8-2\sqrt{2+\sqrt{3}}-2\sqrt{3\left(2-\sqrt{3}\right)}\)

\(\Leftrightarrow\left(\dfrac{8-x_0^2}{2}\right)^2=\left(\sqrt{2+\sqrt{3}}+\sqrt{3\left(2-\sqrt{3}\right)}\right)^2\)

\(=8-2\sqrt{3}+2\sqrt{3}=8\)

\(\Rightarrow x_0^4-16x_0^2+64=32\)

\(\Rightarrow x_0^4-16x_0^2+32=0\)

Vậy ......

Đặt \(x_0=\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\).Ta sẽ chứng minh x₀ là nghiệm của phương trình \(x^4-16x^2+32=0\)

Ta có: \(x_0=\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\)

\(\Rightarrow x_0^2=2+\sqrt{2+\sqrt{3}}+6-3\sqrt{2+\sqrt{3}}\)

\(-2\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{6-3\sqrt{2+\sqrt{3}}}\)

\(=8-2\sqrt{2+\sqrt{3}}-2\sqrt{3}.\sqrt{4-\left(2+\sqrt{3}\right)}\)

\(=8-2\sqrt{2+\sqrt{3}}-2\sqrt{3\left(2-\sqrt{3}\right)}\)

\(\Rightarrow x_0^2-8=-2\sqrt{2+\sqrt{3}}-2\sqrt{3\left(2-\sqrt{3}\right)}\)

\(\Rightarrow\left(x_0^2-8\right)^2=\left[-2\sqrt{2+\sqrt{3}}-2\sqrt{3\left(2-\sqrt{3}\right)}\right]^2\)

\(\Leftrightarrow x_0^4-16x_0^2+64=4\left(2+\sqrt{3}\right)+12\left(2-\sqrt{3}\right)+8\sqrt{3}\)

\(\Leftrightarrow x_0^4-16x_0^2+64=32\)

\(\Leftrightarrow x_0^4-16x_0^2+32=0\)

Điều này chứng tỏ x₀ là nghiệm của phương trình \(x^4-16x^2+32=0\)

Vậy \(x_0\sqrt{2+\sqrt{2+\sqrt{3}}}-\sqrt{6-3\sqrt{2+\sqrt{3}}}\)là nghiệm của phương trình \(x^4-16x^2+32=0\)(đpcm)

\(x^2=8-2\sqrt{2+\sqrt{3}}-2\sqrt{3.\left(2-\sqrt{3}\right)}\)

\(\Leftrightarrow8-x^2=2\sqrt{2+\sqrt{3}}+2\sqrt{3.\left(2-\sqrt{3}\right)}\)

\(\Leftrightarrow x^4-16x^2+64=4\left(2+\sqrt{3}+6-3\sqrt{3}+2\sqrt{3}\right)\)

\(\Leftrightarrow x^4-16x^2+64=32\)

\(\Leftrightarrow x^4-16x^2+32=0\)

Vậy có điều phải chứng minh.