\(\left(2x+1\right)^2-\left(2x-3\right)\left(2x+3\right)=14\\ \Leftrightarrow4x^2+4x+1-4x^2+9=14\\ \Leftrightarrow4x=4\\ \Leftrightarrow x=1\)

a: \(=\dfrac{2\left(x+2\right)\left(x-1\right)}{x+2}=2x-2\)

b: \(=\dfrac{2x^3+x^2-6x^2-3x+2x+1}{2x+1}=x^2-3x+1\)

c: \(=\dfrac{x^3+2x^2-2x^2-4x+2x+4}{x+2}=x^2-2x+2\)

d: \(=\dfrac{x^2\left(x-3\right)}{x-3}=x^2\)

`@` `\text {Ans}`

`\downarrow`

`#3107.\text {DN01012007}`

\(\left(x-5\right)\cdot\left(3-x\right)=0\\ \Rightarrow\left[{}\begin{matrix}x-5=0\\3-x=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=0+5\\x=3-0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=5\\x=3\end{matrix}\right.\)

Vậy, \(x\in\left\{3;5\right\}\)

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\(\left(2x-8\right)\cdot\left(5-x\right)=0\\ \Rightarrow\left[{}\begin{matrix}2x-8=0\\5-x=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x=8\\x=5\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=8\div2\\x=5\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=4\\x=5\end{matrix}\right.\)

Vậy, \(x\in\left\{4;5\right\}\)

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\(7x\left(2x-14\right)=0\\ \Rightarrow\left[{}\begin{matrix}7x=0\\2x-14=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=0\\2x=14\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=0\\x=14\div2\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=0\\x=7\end{matrix}\right.\)

Vậy, \(x\in\left\{0;7\right\}\)

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\(\left(2x-4\right)\cdot\left(6-2x\right)=0\\ \Rightarrow\left[{}\begin{matrix}2x-4=0\\6-2x=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}2x=4\\2x=6\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=4\div2\\x=6\div2\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=2\\x=3\end{matrix}\right.\)

Vậy, \(x\in\left\{2;3\right\}.\)

a/ ĐKXĐ: \(x\ge1\)

Khi \(x\ge1\) ta thấy \(\left\{{}\begin{matrix}VT>0\\VP=1-x\le0\end{matrix}\right.\) nên pt vô nghiệm

b/ \(x\ge1\)

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

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

Đặt \(\sqrt{x-1}=a\ge0\) ta được:

\(\sqrt{a\left(a-1\right)^2}+\sqrt{a\left(a-2\right)^2}=a\)

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

\(\Leftrightarrow\left|a-1\right|+\left|a-2\right|=\sqrt{a}\)

- Với \(a\ge2\) ta được: \(2a-3=\sqrt{a}\Leftrightarrow2a-\sqrt{a}-3=0\Rightarrow\left[{}\begin{matrix}\sqrt{a}=-1\left(l\right)\\\sqrt{a}=\frac{3}{2}\end{matrix}\right.\)

\(\Rightarrow a=\frac{9}{4}\Rightarrow\sqrt{x-1}=\frac{9}{4}\Rightarrow...\)

- Với \(0\le a\le1\) ta được:

\(1-a+2-a=\sqrt{a}\Leftrightarrow2a+\sqrt{a}-3=0\Rightarrow\left[{}\begin{matrix}a=1\\a=-\frac{3}{2}\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\sqrt{x-1}=1\Rightarrow...\)

- Với \(1< a< 2\Rightarrow a-1+2-a=\sqrt{a}\Leftrightarrow a=1\left(l\right)\)

c/ ĐKXĐ: \(x\ge\frac{49}{14}\)

\(\Leftrightarrow\sqrt{14x-49+14\sqrt{14x-49}+49}+\sqrt{14x-49-14\sqrt{14x-49}+49}=14\)

\(\Leftrightarrow\sqrt{\left(\sqrt{14x-49}+7\right)^2}+\sqrt{\left(\sqrt{14x-49}-7\right)^2}=14\)

\(\Leftrightarrow\left|\sqrt{14x-49}+7\right|+\left|7-\sqrt{14x-49}\right|=14\)

Mà \(VT\ge\left|\sqrt{14x-49}+7+7-\sqrt{14x-49}\right|=14\)

Nên dấu "=" xảy ra khi và chỉ khi:

\(7-\sqrt{14x-49}\ge0\)

\(\Leftrightarrow14x-49\le49\Leftrightarrow x\le7\)

Vậy nghiệm của pt là \(\frac{49}{14}\le x\le7\)

d/ ĐKXĐ: \(x\ge\frac{1}{2}\)

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

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

TH1: \(\sqrt{2x-1}\ge3\Rightarrow x\ge5\)

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

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

\(\Leftrightarrow x=13\)

TH2: \(2\le\sqrt{2x-1}< 3\Rightarrow\frac{5}{2}\le x< 5\)

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

\(\Leftrightarrow\sqrt{2x-1}=2\Rightarrow x=\frac{5}{2}\)

TH3: \(1\le\sqrt{2x-1}< 2\Rightarrow1\le x< \frac{5}{2}\)

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

\(\Leftrightarrow4=4\) (luôn đúng)

TH4: \(\frac{1}{2}\le x< 1\)

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

\(\Leftrightarrow\sqrt{2x-1}=1\Rightarrow x=1\left(l\right)\)

Vậy nghiệm của pt là: \(\left[{}\begin{matrix}1\le x\le\frac{5}{2}\\x=13\end{matrix}\right.\)

a: Ta có: \(3\left|2x+5\right|\ge0\forall x\)

\(\Leftrightarrow3\left|2x+5\right|-7\ge-7\forall x\)

Dấu '=' xảy ra khi \(x=-\dfrac{5}{2}\)

c: ta có: \(\left(2x-3\right)^2\ge0\forall x\)

\(\Leftrightarrow\left(2x-3\right)^2-14\ge-14\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{3}{2}\)