ĐK: \(x^3+3x^2-3x+1\ge0\)

\(pt\Leftrightarrow\sqrt[3]{9x^2-15x+9}-\left(2-x\right)+\sqrt{x^3+3x^2-3x+1}=0\)

\(\Leftrightarrow\frac{9x^2-15x+9-\left(2-x\right)^3}{A^2+AB+B^2}+\sqrt{x^3+3x^2-3x+1}=0\)

\(\left(A=\sqrt[3]{9x^2-15x+9};\text{ }B=2-x\right)\)\(\text{(}A^2+AB+B^2=\left(A+\frac{B}{2}\right)^2+\frac{3B^2}{4}>0\text{)}\)

\(\Leftrightarrow\frac{x^3+3x^2-3x+1}{A^2+AB+B^2}+\sqrt{x^3+3x^2-3x+1}=0\)

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

\(\Leftrightarrow x^3+3x^2-3x+1=0\text{ (do }\frac{\sqrt{x^3+3x^2-3x+1}}{A^2+AB+B^2}+1>0\text{)}\)

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

\(\Leftrightarrow x+1+\sqrt[3]{2}+\sqrt[3]{4}=0\text{ (}pt\text{ }x^2+\left(2-\sqrt[3]{2}-\sqrt[3]{4}\right)x+\sqrt[3]{2}-1=0\text{ vô nghiệm do }\Delta

ĐKXĐ: \(x\ge\dfrac{1}{3}\)

\(\Leftrightarrow x^2+11x-3+2\sqrt{\left(x^2+2x\right)\left(9x-3\right)}=4x^2+13x+3\)

\(\Leftrightarrow2\sqrt{\left(x^2+2x\right)\left(9x-3\right)}=3x^2+2x+6\)

\(\Leftrightarrow2\sqrt{\left(3x+6\right)\left(3x^2-x\right)}=3x^2+2x+6\)

\(\Leftrightarrow\left(3x^2-x\right)-2\sqrt{\left(3x+6\right)\left(3x^2-x\right)}+3x+6=0\)

\(\Leftrightarrow\left(\sqrt{3x^2-x}-\sqrt{3x+6}\right)^2=0\)

\(\Leftrightarrow3x^2-x=3x+6\)

\(\Leftrightarrow3x^2-4x-6=0\Rightarrow\left[{}\begin{matrix}x=\dfrac{2+\sqrt{22}}{3}\\x=\dfrac{2-\sqrt{22}}{3}\left(loại\right)\end{matrix}\right.\)

Giải phương trình $x^2-4x+6=\sqrt{2x^2-5x+3}+\sqrt{-3x^2+9x-5}$ - Phương trình - hệ phương trình - bất phương trình - Diễn đàn Toán học

hình như bạn ghi sai đề phải k ạ 

Sai đề :D

\(ĐK:\left\{{}\begin{matrix}x\le\dfrac{1}{2};4\le x\\\dfrac{1}{2}\le x\\x\le-11;\dfrac{1}{2}\le x\end{matrix}\right.\Leftrightarrow x\le-11;4\le x\)

\(PT\Leftrightarrow\sqrt{\left(x-4\right)\left(2x-1\right)}+3\sqrt{2x-1}-\sqrt{\left(2x-1\right)\left(x+11\right)}=0\\ \Leftrightarrow\sqrt{2x-1}\left(\sqrt{x-4}-\sqrt{x+11}+3\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}2x-1=0\\\sqrt{x-4}-\sqrt{x+11}=-3\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}\left(tm\right)\\x-4+x+11-2\sqrt{x^2+7x-44}=9\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow2\sqrt{x^2+7x-44}=2x-2\\ \Leftrightarrow\sqrt{x^2+7x-44}=x-1\\ \Leftrightarrow x^2+7x-44=x^2-2x+1\\ \Leftrightarrow9x=45\Leftrightarrow x=5\left(tm\right)\)

Vậy \(S=\left\{\dfrac{1}{2};5\right\}\)

https://hoc24.vn/cau-hoi/giai-pt-sqrt2x2-9x43sqrt2x-1sqrt2x221x-11.2005877637936

làm r nha :vv

c.

ĐKXĐ: \(\left[{}\begin{matrix}x\le-5\\x\ge6\end{matrix}\right.\)

\(\sqrt{\left(x-3\right)\left(x-5\right)}+\sqrt{\left(x-3\right)\left(x+5\right)}=\sqrt{\left(x-3\right)\left(x-6\right)}\)

- Với \(x\ge6\) , do \(x-3>0\) pt trở thành:

\(\sqrt{x-5}+\sqrt{x+5}=\sqrt{x-6}\)

Do \(\left\{{}\begin{matrix}\sqrt{x-5}>\sqrt{x-6}\\\sqrt{x+5}>0\end{matrix}\right.\) \(\Rightarrow\sqrt{x-5}+\sqrt{x+5}>\sqrt{x-6}\) pt vô nghiệm

- Với \(x\le-5\) pt tương đương:

\(\sqrt{\left(3-x\right)\left(5-x\right)}+\sqrt{\left(3-x\right)\left(-x-5\right)}=\sqrt{\left(3-x\right)\left(6-x\right)}\)

Do \(3-x>0\) pt trở thành:

\(\sqrt{5-x}+\sqrt{-x-5}=\sqrt{6-x}\)

\(\Leftrightarrow-2x+2\sqrt{x^2-25}=6-x\)

\(\Leftrightarrow2\sqrt{x^2-25}=x+6\) (\(x\ge-6\))

\(\Leftrightarrow4\left(x^2-25\right)=x^2+12x+36\)

\(\Leftrightarrow3x^2-12x-136=0\Rightarrow x=\dfrac{6-2\sqrt{111}}{3}\)

a.

Kiểm tra lại đề, pt này không giải được

b.

ĐKXĐ: \(x\ge0\)

\(\sqrt{x\left(x+1\right)}-\sqrt{x}+1-\sqrt{x+1}=0\)

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

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

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x}=1\\\sqrt{x+1}=1\end{matrix}\right.\)

\(\Leftrightarrow...\)