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TXĐ: \(x\ge0\)

Phương trình đã cho tương đương:

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

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

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

\(\Leftrightarrow x-1=0\) (do \(1+\dfrac{1}{\sqrt{2x+1}+\sqrt{3x}}>0\) \(\forall x\ge0\))

\(\Leftrightarrow x=1\)

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

Điều kiện : x\(\ge0\)

\(\Leftrightarrow\sqrt{2x+1}=x-1+\sqrt{3x}\)

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

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

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

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

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

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

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

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

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

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=0\\x\in\varnothing\end{matrix}\right.\) Vậy pt tập nghiệm S={1;0}

Pt tương đương:

\(\sqrt[3]{4x-3}\)-\(\sqrt[3]{3x+1}\)=\(\sqrt[3]{5-x}\)+\(\sqrt[3]{2x-9}\)

\(\Leftrightarrow\)-3\(\sqrt[3]{\text{(4x-3)(3x+1)}}\)(\(\sqrt[3]{4x-3}\)-\(\sqrt[3]{3x+1}\))=3\(\sqrt[3]{\left(5-x\right)\left(2x-9\right)}\)(\(\sqrt[3]{5-x}\)+\(\sqrt[3]{2x-9}\))

\(\Leftrightarrow\)\(\orbr{\begin{cases}\sqrt[3]{4x-3}-\sqrt[3]{3x+1}=\sqrt[3]{5-x}+\sqrt[3]{2x-9}=0\left(1\right)\\3\sqrt[3]{-12x^2+5x+3}=3\sqrt[3]{-2x^2+19x-45}\left(2\right)\end{cases}}\)

(1)<=>4x-3=3x+1 và x-5=2x-9<=>x=4

(2)<=>-12x²+5x+3=-2x²+19x-45<=>-5x²-7x+24=0<=>x=8/5 và x=-3

 bạn thử các giá trị x=4,x=8/5 và x=-3 vào pt và kết luận

mik ko hieu vi sao ban suy ra duoc (1) va (2)

bn co the viet ro ra duoc ko ?

theo mik thay thi 2 pt do dau co tuong duong

Mình chuyển vế rồi lập phương, do  4x-3-(3x+1)=2x-9+(5-x) nên mình giản bỏ luôn, hơi tắc xíu

ĐKXĐ: \(x\ge-1\)

Đặt \(\sqrt{x+1}=y\ge0\)

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

\(\Leftrightarrow x^2-3xy+2y^2=0\Leftrightarrow x^2-xy-2xy+2y^2=0\Leftrightarrow x\left(x-y\right)-2y\left(x-y\right)=0\)

\(\Leftrightarrow\left(x-2y\right)\left(x-y\right)=0\Leftrightarrow\orbr{\begin{cases}x=2y\\x=y\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=2\sqrt{x+1}\\x=\sqrt{x+1}\end{cases}}\)

Đến đây đơn giản rồi bạn giải từng trường hợp là ra

DDK : \(x\ge1\)

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

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

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

\(\Leftrightarrow x-1-3x+2-5x+2=2\sqrt{15x^2-3x-10x+2}\)

\(\Leftrightarrow3-7x=2\sqrt{15x^2-13x+2}\)

\(\Rightarrow9-42x+49x^2=4\left(15x^2-13x+2\right)\)

\(\Leftrightarrow9-42x+49x^2=60x^2-52x+8\)

\(\Leftrightarrow11x^2-10x-1=0\)

\(\Leftrightarrow11x^2-11x+x-1=0\)

\(\Leftrightarrow\left(11x+1\right)\left(x-1\right)=0\)

Giải nốt nha .

Lời giải:

Với mọi $x$ thuộc ĐKXĐ, ta luôn có:

\(\left\{\begin{matrix} \sqrt{3x+x^2+\frac{9}{4}}\geq 0\\ \sqrt{x^2+3x+1}\geq 0\end{matrix}\right.\)

Do đó, để \(\sqrt{3x+x^2+\frac{9}{4}}+\sqrt{x^2+3x+1}=0\) thì:

\(\left\{\begin{matrix} \sqrt{3x+x^2+\frac{9}{4}}= 0\\ \sqrt{x^2+3x+1}=0\end{matrix}\right.\)

\(\Leftrightarrow \left\{\begin{matrix} x=\frac{-3}{2}\\ x=\frac{3\pm \sqrt{5}}{2}\end{matrix}\right.\) (vô lý)

Do đó pt vô nghiệm.

mik ko biết vì mới chỉ học lớp 6

ĐKXĐ: \(x\ge\frac{1}{2}\)

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

Nhân liên hợp ta được:

\(\frac{\left(\sqrt{\frac{x+7}{x+1}}-\sqrt{3}\right)\left(\sqrt{\frac{x+7}{x+1}}+\sqrt{3}\right)}{\sqrt{\frac{x+7}{x+1}}+\sqrt{3}}+2\left(4-x^2\right)-\frac{\left(\sqrt{2x-1}-\sqrt{3}\right)\left(\sqrt{2x+1}+\sqrt{3}\right)}{\sqrt{2x+1}+\sqrt{3}}=0\)

\(\Rightarrow\frac{\frac{x+7}{x+1}-3}{\sqrt{\frac{x+7}{x+1}}+\sqrt{3}}+2\left(4-x^2\right)-\frac{2x-1-3}{\sqrt{2x+1}+\sqrt{3}}=0\)

\(\Rightarrow\frac{\frac{-2x+4}{x+1}}{\sqrt{\frac{x+7}{x+1}}+\sqrt{3}}+2\left(2-x\right)\left(2+x\right)-\frac{2x-4}{\sqrt{2x+1}+\sqrt{3}}=0\)

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

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

=> x - 2 = 0 => x = 2

                                                   Vậy x = 2

rảnh  quá