Tham khảo:

1) Giải phương trình : \(11\sqrt{5-x}+8\sqrt{2x-1}=24+3\sqrt{\left(5-x\right)\left(2x-1\right)}\) - Hoc24

\(\sqrt{x+3}+2\sqrt{x}=2+\sqrt{x\left(x+3\right)}\left(đk:x\ge0\right)\)

\(\Leftrightarrow x+3+4x+4\sqrt{x\left(x+3\right)}=4+x\left(x+3\right)+4\sqrt{x\left(x+3\right)}\)

\(\Leftrightarrow5x+3=4+x^2+3x\)

\(\Leftrightarrow x^2-2x+1=0\)

\(\Leftrightarrow\left(x-1\right)^2=0\)

\(\Leftrightarrow x-1=0\)

\(\Leftrightarrow x=1\left(tm\right)\)

\(11\sqrt{5-x}+8\sqrt{2x-1}=24+3\sqrt{\left(5-x\right)\left(2x-1\right)}\)

\(\Leftrightarrow11\sqrt{5-x}+8\sqrt{2x-1}=24+3\sqrt{11x-5-2x^2}\)

\(\Leftrightarrow121\left(5-x\right)+176\sqrt{\left(5-x\right)\left(2x-1\right)}+64\left(2x-1\right)=576+144\sqrt{11x-5-2x^2}\)\(+9\left(11x-5-2x^2\right)\)

\(\Leftrightarrow605-121x+176\sqrt{11x-5-2x^2}+128x-64=576+144\sqrt{11x-5-2x^2}\)\(+99x-18x^2\)

\(\Leftrightarrow176\sqrt{11x-5-2x^2}-144\sqrt{11x-5-2x^2}=531+99x-18x^2-541-7x\)

\(\Leftrightarrow32\sqrt{11x-5-2x^2}=-10+92x-18x^2\)

\(\Leftrightarrow16\sqrt{11x-5-2x^2}=-5+46x-9x^2\)

\(\Leftrightarrow256\left(11x-5-2x^2\right)=25+2116x^2+81x^4-460x+90x^2-823x^3\)

\(\Leftrightarrow2816x-1280-512x^2=25+2206x^2+81x^4-460x-823x^3\)

\(\Leftrightarrow9\left(364x-145-302x^2-9x^4+92x^3\right)=0\)

\(\Leftrightarrow-9x^4+92x^3-302x^2+364x-145=0\)

\(\Leftrightarrow-\left(x-1\right)\left(9x^3-83x^2+219x-145\right)=0\)

\(\Leftrightarrow-\left(x-1\right)\left(x-1\right)\left(9x^2-74x+145\right)=0\)

\(\Leftrightarrow-\left(x-1\right)^2\left(9x-29\right)\left(x-5\right)=0\Leftrightarrow\)x=1; x=29_/9; x=5

\(\Leftrightarrow11\sqrt{5-x}+8\sqrt{2x-1}=24+3\sqrt{11x-5-2x^2}\)

1.

\(\Leftrightarrow\left(2x+1\right)\sqrt{2x^2+4x+5}-\left(2x+1\right)\left(x+3\right)+x^2-2x-4=0\)

\(\Leftrightarrow\left(2x+1\right)\left(\sqrt{2x^2+4x+5}-\left(x+3\right)\right)+x^2-2x-4=0\)

\(\Leftrightarrow\dfrac{\left(2x+1\right)\left(x^2-2x-4\right)}{\sqrt{2x^2+4x+5}+x+3}+x^2-2x-4=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2-2x-4=0\\\dfrac{2x+1}{\sqrt{2x^2+4x+5}+x+3}+1=0\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow2x+1+\sqrt{2x^2+4x+5}+x+3=0\)

\(\Leftrightarrow\sqrt{2x^2+4x+5}=-3x-4\) \(\left(x\le-\dfrac{4}{3}\right)\)

\(\Leftrightarrow2x^2+4x+5=9x^2+24x+16\)

\(\Leftrightarrow7x^2+20x+11=0\)

2.

ĐKXĐ: ...

\(\Leftrightarrow2x\sqrt{2x+7}+7\sqrt{2x+7}=x^2+2x+7+7x\)

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

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

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

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

\(\Leftrightarrow...\)

3.

ĐKXĐ: ...

Từ pt dưới:

\(\Leftrightarrow\left(x-y\right)\left(x^2+xy+y^2\right)+3x-3y=3x^2+3y^2+1+1\)

\(\Leftrightarrow x^3-y^3+3x-3y=3x^2+3y^2+1+1\)

\(\Leftrightarrow x^3-3x^2+3x-1=y^3+3y^2+3y+1\)

\(\Leftrightarrow\left(x-1\right)^3=\left(y+1\right)^3\)

\(\Leftrightarrow y=x-2\)

Thế vào pt trên:

\(x^2-2x+3=2\sqrt{5x-2}+\sqrt{7x-1}\)

\(\Leftrightarrow x^2-5x+2+2\left(x-\sqrt{5x-2}\right)+\left(x+1-\sqrt{7x-1}\right)=0\)

\(\Leftrightarrow x^2-5x+2+\dfrac{2\left(x^2-5x+2\right)}{x+\sqrt{5x-2}}+\dfrac{x^2-5x+2}{x+1+\sqrt{7x-1}}=0\)

\(\Leftrightarrow x^2-5x+2=0\)

kuchiyose edo tensei

nhờ vào năng lực rinegan , ta có thể  đoán dc

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

vậy pt sẽ như sau

\(a,\left(\sqrt{1+x}+\sqrt{8-x}\right)^2-\sqrt{\left(1+x\right)\left(8-x\right)}=3\) " thêm bớt nếu m thông minh sẽ hiểu "

\(9+2\sqrt{\left(1+x\right)\left(8-x\right)}-\sqrt{\left(1+x\right)\left(8-x\right)}=3\)

\(\sqrt{\left(1+x\right)\left(8-x\right)}=-6\)

\(\left(1+x\right)\left(8-x\right)=36\)

đến đây m có thể tự làm

c)  \(\sqrt{x+5}=5-x^2\)

      \(x+5=\left(5-x\right)^2\)

     \(x+5=x^4-10x^2+25\)  " rồi xong pt bậc 4 :)

 \(x^4-10x^2-x+20=0\)

\(x^4=10x^2+x-20\)

\(x^4+2mx^2+m^2=10x^2+x-20+2mx^2+m^2\)

\(\left(x^2+m\right)^2=2x^2\left(5+m\right)+x+\left(m^2-20\right)\)

\(\Delta=1-8\left(5+m\right)\left(m^2-20\right)\)

\(\Delta=1-8\left(5m^2-100+m^3-20m\right)\)

\(\Delta=1-40m^2+800-8m^3+160m\)

\(\Delta=-\left(2m+9\right)\left(4m^2+2m-89\right)\)

lấy m= -9/2 , cho nhanh thay vào ta đươc

\(\left(x^2-\frac{9}{2}\right)^2=2x^2\left(5-\frac{9}{2}\right)+x+\left(\frac{9}{2}^2-20\right)\)

\(\left(x^2-\frac{9}{2}\right)^2=x^2+x+\frac{1}{4}\)

\(\left(x^2-\frac{9}{2}\right)^2=\left(x+\frac{1}{2}\right)^2\)

\(\hept{\begin{cases}x^2-\frac{9}{2}=x+\frac{1}{2}\\x^2-\frac{9}{2}=-x-\frac{1}{2}\end{cases}}\)

đến đây cậu có thể làm tiếp :)

câu B hơi gắt cần time suy nghĩ :)

sửa lại dòng đầu tiên phải là + 2 căn (1+x)(8-x) nhé :)

a) đkxđ \(x\ge1\)

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

\(\Leftrightarrow\dfrac{2x-10}{\sqrt{2x-1}+3}+\dfrac{x-5}{\sqrt{x-1}+2}=0\)

\(\Leftrightarrow\left(x-5\right)\left(\dfrac{2}{\sqrt{2x-1}+3}+\dfrac{1}{\sqrt{x-1}+2}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=5\left(nhận\right)\\\dfrac{2}{\sqrt{2x-1}+3}+\dfrac{1}{\sqrt{x-1}+3}=0\end{matrix}\right.\)

 Hiển nhiên pt thứ 2 vô nghiệm vì \(VT>0\) với mọi \(x\ge1\). Do đó pt đã cho có nghiệm duy nhất là \(x=5\)

b) đkxđ: \(x\ge-3\)

 Để ý rằng \(x^2+2x+7=\left(x^2+1\right)+\left(2x+6\right)=\left(x^2+1\right)+2\left(x+3\right)\) nên nếu ta đặt \(\sqrt{x^2+1}=u\left(u\ge1\right)\) và \(\sqrt{x+3}=v\left(v\ge0\right)\) thì pt đã chot rở thành:

 \(u^2+2v^2=3uv\)

 \(\Leftrightarrow\left(u-v\right)\left(u-2v\right)=0\)

 \(\Leftrightarrow\left[{}\begin{matrix}u=v\\u=2v\end{matrix}\right.\)

Nếu \(u=v\) thì \(\sqrt{x^2+1}=\sqrt{x+3}\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge-3\\x^2+1=x+3\end{matrix}\right.\) 

Mà \(x^2+1=x+3\)  \(\Leftrightarrow x^2-x-2=0\)

\(\Leftrightarrow\left(x+1\right)\left(x-2\right)=0\) \(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-1\end{matrix}\right.\) (nhận)

 Nếu \(u=2v\) thì \(\sqrt{x^2+1}=2\sqrt{x+3}\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-3\\x^2+1=4x+12\end{matrix}\right.\)

mà \(x^2+1=4x+12\)\(\Leftrightarrow x^2-4x-11=0\)

\(\Leftrightarrow x=2\pm\sqrt{15}\) (nhận)

Vậy pt đã cho có tập nghiệm \(S=\left\{2;-1;2\pm\sqrt{15}\right\}\)

a) \(\sqrt{2x-1}+\sqrt{x-1}=5\) (ĐK: \(x\ge1\)) 

\(\Leftrightarrow\left(\sqrt{2x-1}+\sqrt{x-1}\right)^2=5^2\)

\(\Leftrightarrow2x-1+x-1+2\sqrt{\left(2x-1\right)\left(x-1\right)}=25\)

\(\Leftrightarrow3x-2+2\sqrt{\left(2x-1\right)\left(x-1\right)}=25\)

\(\Leftrightarrow\sqrt{\left(2x-1\right)\left(x-1\right)}=\dfrac{27-3x}{2}\)

\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{27-3x}{2}\ge0\\\left(2x-1\right)\left(x-1\right)=\left(\dfrac{27-3x}{2}\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}27-3x\ge0\\2x^2-2x-x+1=\dfrac{729-162x+9x^2}{4}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}3x\le27\\8x^2-12x+4=9x^2-162x+729\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le9\\x^2-150x+725=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le9\\\left[{}\begin{matrix}x-5=0\\x-145=0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le9\\\left[{}\begin{matrix}x=5\left(tm\right)\\x=145\left(ktm\right)\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow x=5\)