Đk: \(x\ge\frac{1}{2}\)

\(PT\Leftrightarrow2x^3+8+\left(\sqrt{2x-1}-\sqrt{\frac{x+7}{x+1}}\right)=0\)

\(\Leftrightarrow2x^2-8+\frac{2x-1-\frac{x+7}{x+1}}{\sqrt{2x-1}+\sqrt{\frac{x+7}{x+1}}}=0\)

\(\Leftrightarrow2x^2-8+\frac{2x^2-8}{\left(x+1\right)\left(\sqrt{2x-1}+\sqrt{\frac{x+7}{x+1}}\right)}=0\)

\(\Leftrightarrow\left(2x^2-8\right)\left[1+\frac{1}{\left(x+1\right)\left(\sqrt{2x-1}+\sqrt{\frac{x+7}{x+1}}\right)}\right]=0\)

\(\Rightarrow x=2\) (phần trong ngoặc luôn dương)

Đổi k ko minasan

Câu 2/

Điều kiện xác định b tự làm nhé:

\(\frac{6}{x^2-9}+\frac{4}{x^2-11}-\frac{7}{x^2-8}-\frac{3}{x^2-12}=0\)

\(\Leftrightarrow x^4-25x^2+150=0\)

\(\Leftrightarrow\left(x^2-10\right)\left(x^2-15\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x^2=10\\x^2=15\end{cases}}\)

Tới đây b làm tiếp nhé.

a. ĐK: \(\frac{2x-1}{y+2}\ge0\)

Áp dụng bđt Cô-si ta có: \(\sqrt{\frac{y+2}{2x-1}}+\sqrt{\frac{2x-1}{y+2}}\ge2\)

\(\)Dấu bằng xảy ra khi  \(\frac{y+2}{2x-1}=1\Rightarrow y+2=2x-1\Rightarrow y=2x-3\) 

Kết hợp với pt (1) ta tìm được x = -1, y = -5 (tmđk)

b. \(pt\Leftrightarrow\left(\frac{6}{x^2-9}-1\right)+\left(\frac{4}{x^2-11}-1\right)-\left(\frac{7}{x^2-8}-1\right)-\left(\frac{3}{x^2-12}-1\right)=0\)

\(\Leftrightarrow\left(15-x^2\right)\left(\frac{1}{x^2-9}+\frac{1}{x^2-11}+\frac{1}{x^2-8}+\frac{1}{x^2-12}\right)=0\)

\(\Leftrightarrow x^2-15=0\Leftrightarrow\orbr{\begin{cases}x=\sqrt{15}\\x=-\sqrt{15}\end{cases}}\)

BALABOLO

TK NHA

Nhìn không đủ chán rồi không dám động vào

Viết đề kiểu gì v @@

À do nãy máy lag sr :) Chứ bài đặt ẩn phụ mệt lắm :)

c.

\(\Leftrightarrow x^2+3-\left(3x+1\right)\sqrt{x^2+3}+2x^2+2x=0\)

Đặt \(\sqrt{x^2+3}=t>0\)

\(\Rightarrow t^2-\left(3x+1\right)t+2x^2+2x=0\)

\(\Delta=\left(3x+1\right)^2-4\left(2x^2+2x\right)=\left(x-1\right)^2\)

\(\Rightarrow\left[{}\begin{matrix}t=\dfrac{3x+1-x+1}{2}=x+1\\t=\dfrac{3x+1+x-1}{2}=2x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+3}=x+1\left(x\ge-1\right)\\\sqrt{x^2+3}=2x\left(x\ge0\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2+3=x^2+2x+1\left(x\ge-1\right)\\x^2+3=4x^2\left(x\ge0\right)\end{matrix}\right.\)

\(\Leftrightarrow x=1\)

a.

Đề bài ko chính xác, pt này ko giải được

b.

ĐKXĐ: \(x\ge-\dfrac{7}{2}\)

\(2x+7-\left(2x+7\right)\sqrt{2x+7}+x^2+7x=0\)

Đặt \(\sqrt{2x+7}=t\ge0\)

\(\Rightarrow t^2-\left(2x+7\right)t+x^2+7x=0\)

\(\Delta=\left(2x+7\right)^2-4\left(x^2+7x\right)=49\)

\(\Rightarrow\left[{}\begin{matrix}t=\dfrac{2x+7-7}{2}=x\\t=\dfrac{2x+7+7}{2}=x+7\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{2x+7}=x\left(x\ge0\right)\\\sqrt{2x+7}=x+7\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2-2x-7=0\left(x\ge0\right)\\x^2+12x+42=0\left(vn\right)\end{matrix}\right.\)

\(\Rightarrow x=1+2\sqrt{2}\)

\(\sqrt{\sqrt{2}-1-x}+\sqrt[4]{x}=\frac{1}{\sqrt[4]{2}}\)

ĐKXĐ: Tự tìm nhé.

\(\left(\sqrt{\sqrt{2}-1-x};\sqrt[4]{x}\right)\rightarrow\left(b;a\right)\)

Phương trình <=>  \(\hept{\begin{cases}a+b=\frac{1}{\sqrt[4]{2}}\\a^4+b^2=\sqrt{2}-1\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}b=\frac{1}{\sqrt[4]{2}}-a\\a^4+b^2=\sqrt{2}-1\left(2\right)\end{cases}}\)

(2) <=> \(a^4+a^2-\frac{2}{\sqrt[4]{2}}a+\frac{1}{\sqrt{2}}-\sqrt{2}+1=0\)

\(\Leftrightarrow\sqrt{2}a^4+\sqrt{2}a^2-2\sqrt[4]{2}a+\sqrt{2}-1=0\)

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

\(\Leftrightarrow a^2-a+\frac{\sqrt{2}-\sqrt[4]{2}}{\sqrt{2}}=0\)( vì \(\Leftrightarrow\sqrt{2}a^2+\sqrt{2}a+2\sqrt{2}+\sqrt[4]{2}-\sqrt{2}>0\))

Tự làm tiếp nhé

ĐK: \(x\ge\frac{1}{2}\)

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

\(\Leftrightarrow\left(\sqrt{\frac{x+7}{x+1}}-\sqrt{3}\right)+2\left(2-x\right)\left(2+x\right)=\left(\sqrt{2x-1}-\sqrt{3}\right)\)

\(\Leftrightarrow\frac{2\left(2-x\right)}{\sqrt{\left(x+7\right)\left(x+1\right)}+\sqrt{3}\left(x+1\right)}+2\left(2-x\right)\left(2+x\right)=\frac{2\left(x-2\right)}{\sqrt{2x-1}+\sqrt{3}}\)

\(\Leftrightarrow\frac{2\left(2-x\right)}{\sqrt{\left(x+7\right)\left(x+1\right)}+\sqrt{3}\left(x+1\right)}+2\left(2-x\right)\left(2+x\right)+\frac{2\left(2-x\right)}{\sqrt{2x-1}+\sqrt{3}}=0\)

\(\Leftrightarrow\left(2-x\right)\left[\frac{2}{\sqrt{\left(x+7\right)\left(x+1\right)}+\sqrt{3}\left(x+1\right)}+2\sqrt{2+x}+\frac{2}{\sqrt{2x-1}+\sqrt{3}}\right]=0\)

\(\Leftrightarrow x=2\)( \(\frac{2}{\sqrt{\left(x+7\right)\left(x+1\right)}+\sqrt{3}\left(x+1\right)}+2\left(2+x\right)+\frac{2}{\sqrt{2x-1}+\sqrt{3}}>0\))

KL:...