ĐK: \(x^2-1\ge0\)

pt <=> \(\left(x^2+2x+1\right)-2\left(x+1\right)\sqrt{x^2-1}+\left(x^2-1\right)-4x^2+4x-1=0\)

<=> \(\left[\left(x+1\right)^2-2\left(x+1\right)\sqrt{x^2-1}+\left(x^2-1\right)\right]-\left(2x-1\right)^2=0\)

<=> \(\left(x+1-\sqrt{x^2-1}\right)^2-\left(2x-1\right)^2=0\)

<=> \(\left(x+1-\sqrt{x^2-1}-2x+1\right)\left(x+1-\sqrt{x^2-1}+2x-1\right)=0\)

Phương trình tích. Dễ rồi đúng ko? Tự làm tiếp nhé!

B = \(\frac{4+\sqrt{7}}{3\sqrt{2}+\sqrt{4+\sqrt{7}}}+\frac{4-\sqrt{7}}{3\sqrt{2}-\sqrt{4-\sqrt{7}}}\)

=>  \(\frac{2}{\sqrt{2}}B=\frac{8+2\sqrt{7}}{6+\sqrt{8+2\sqrt{7}}}+\frac{8-2\sqrt{7}}{6-\sqrt{8-2\sqrt{7}}}\)

=> \(\frac{2}{\sqrt{2}}B=\frac{\left(\sqrt{7}+1\right)^2}{6+\sqrt{7}+1}+\frac{\left(\sqrt{7}-1\right)^2}{6-\sqrt{7}+1}\)

=> \(\frac{2}{\sqrt{2}}B=\frac{\left(\sqrt{7}+1\right)^2}{\sqrt{7}\left(\sqrt{7}+1\right)}+\frac{\left(\sqrt{7}-1\right)^2}{\sqrt{7}\left(\sqrt{7}-1\right)}\)

=> \(\frac{2}{\sqrt{2}}B=\frac{\sqrt{7}+1}{\sqrt{7}}+\frac{\sqrt{7}-1}{\sqrt{7}}=\frac{2\sqrt{7}}{\sqrt{7}}=2\)

=> B = \(\sqrt{2}\)

\(\sqrt{x^2-9}-3\sqrt{x-3}=0\)

\(ĐKXĐ:x\ge3\)

\(pt\Leftrightarrow\sqrt{\left(x+3\right)\left(x-3\right)}-3\sqrt{x-3}=0\)

\(\Leftrightarrow\sqrt{x+3}.\sqrt{x-3}-3.\sqrt{x-3}=0\)

\(\Leftrightarrow\sqrt{x-3}\left(\sqrt{x+3}-3\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}\sqrt{x-3}=0\\\sqrt{x+3}-3=0\end{cases}}\)

\(TH1:\sqrt{x-3}=0\Leftrightarrow x-3=0\Leftrightarrow x=3\left(tm\right)\)

\(TH2:\sqrt{x+3}-3=0\Leftrightarrow\sqrt{x+3}=3\)

\(\Leftrightarrow\sqrt{x+3}=\sqrt{9}\Leftrightarrow x+3=9\Leftrightarrow x=9\left(tm\right)\)

Vậy pt có 2 nghiệm là 3 và 9

Nhầm , 3 và 6 chứ

a,\(x^2-7x+\sqrt{x^2-7x+8}=12\)

ĐKXĐ: .....

Đặt \(x^2-7x=t\)

Phương trình trở thành

\(t+\sqrt{t+8}=12\)

\(\Leftrightarrow\sqrt{t+8}=12-t\)

\(\Leftrightarrow t+8=\left(12-t\right)^2\)

\(\Leftrightarrow t+8=144-24t+t^2\)

\(\Leftrightarrow t^2-25t+136=0\)

\(\Leftrightarrow\left(t-17\right)\left(t-8\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}t-17=0\\t-8=0\end{cases}\Leftrightarrow\orbr{\begin{cases}t=17\\t=8\end{cases}}}\)

tại t = 17 , ta có

\(x^2-7x=17\Leftrightarrow x^2-7x-17=0\)

\(\Leftrightarrow.......\)

Tại t = 8 ta có

\(x^2-7x=8\Leftrightarrow x^2-7x-8=0\)

\(\Leftrightarrow\left(x-8\right)\left(x+1\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x-8=0\\x+1=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=8\\x=-1\end{cases}}}\)

b, \(x^2+4x+5=2\sqrt{2x+3}\)

mik ko bt :)

a,đkxđ:\(x^2-7x+8\ge0\Leftrightarrow x^2-2\cdot\frac{7}{2}x+\frac{49}{4}-\frac{17}{4}\ge0\Leftrightarrow\left(x-\frac{7}{2}\right)^2\ge\frac{17}{4}\Leftrightarrow\hept{\begin{cases}x-\frac{7}{2}\ge\frac{\sqrt{17}}{2}\approx2,06\\x-\frac{7}{2}\le-\frac{\sqrt{17}}{2}\approx-2,06\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge5,56\\x\le1,44\end{cases}}\)

\(\Leftrightarrow\left(x^2-7x+8\right)+\sqrt{x^2-7x+8}=12+8=20\)

\(\Leftrightarrow4\left(x^2-7x+8\right)+4\sqrt{x^2-7x+8}+1=20\cdot4+1=81\)

\(\Leftrightarrow\left(2\sqrt{x^2-7x+8}+1\right)^2=81\)

\(\Leftrightarrow2\sqrt{x^2-7x+8}+1=\pm9\)

Mà vế trái >0 nên \(2\sqrt{x^2-7x+8}+1=9\)

\(\Leftrightarrow\sqrt{x^2-7x+8}=\frac{9-1}{2}=4\)

\(\Leftrightarrow x^2-7x+8=16\)

\(\Leftrightarrow x^2-7x-8=0\Leftrightarrow\left(x-8\right)\left(x+1\right)=0\Leftrightarrow\orbr{\begin{cases}x=8\\x=-1\end{cases}}\)