\(P=\frac{3\left(x+\sqrt{x}-3\right)}{x+\sqrt{x}-2}+\frac{\sqrt{x}+3}{\sqrt{x}+2}-\frac{\sqrt{x}-2}{\sqrt{x}-1}\left(ĐKXĐ:x\ne1;x\ge0\right)\)

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

\(P=\frac{3x+3\sqrt{x}-9}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\frac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\frac{x-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)

\(P=\frac{3x+3\sqrt{x}-9+x+2\sqrt{x}-3-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\)

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

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

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

\(P=\frac{\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}+2\right)}\)

b)Để \(P< \frac{15}{4}\)thì \(\frac{\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}+2\right)}< \frac{15}{4}\)

      Ta có:\(\frac{\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}+2\right)}< \frac{15}{4}\)

          \(\Leftrightarrow\frac{\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}+2\right)}-\frac{15}{4}< 0\)

           \(\Leftrightarrow\frac{12\sqrt{x}+32-15\sqrt{x}-30}{4\left(\sqrt{x}+2\right)}< 0\)

            \(\Leftrightarrow\frac{-\left(3\sqrt{x}+2\right)}{4\sqrt{x}+8}< 0\)

                 Vì \(x\ge0;x\ne1\)

                              Do đó \(0< 4\sqrt{x}+8\)

   Mà \(-\left(3\sqrt{x}+2\right)< 0\)

          Vậy \(P< \frac{15}{4}\left(đpcm\right)\)

c)Ta có:\(P=\frac{\left(3\sqrt{x}+8\right)}{\left(\sqrt{x}+2\right)}\)

             \(\Leftrightarrow P=\frac{3\sqrt{x}+6+2}{\left(\sqrt{x}+2\right)}\)

             \(\Leftrightarrow P=\frac{3\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)}+\frac{2}{2\sqrt{x}+2}\)

              \(\Leftrightarrow P=3+\frac{2}{\sqrt{x}+2}\)

Vì \(x\ge0;x\ne1\Rightarrow\frac{2}{\sqrt{x}+2}\le1\)

       Do đó \(P\le4\Leftrightarrow x=1\)

                Vậy Max P=4 khi x=1

P=3x+3√x−9(√x−1)(√x+2) +√x+3√x+2 −√x−2√x−1 

P=3x+3√x−9(√x−1)(√x+2) +(√x+3)(√x−1)(√x+2)(√x−1) −x−4(√x−1)(√x+2) 

P=3x+3√x−9+x+2√x−3−x+4(√x−1)(√x+2) 

P=3x−8+5√x(√x−1)(√x+2) 

P=3x−3√x+8√x−8(√x−1)(√x+2) 

P=(3√x+8)(√x−1)(√x−1)(√x+2) 

P=(3√x+8)(√x+2) 

b)Để P<154 thì (3√x+8)(√x+2) <154 

      Ta có:(3√x+8)(√x+2) <154 

          ⇔(3√x+8)(√x+2) −154 <0

           ⇔12√x+32−15√x−304(√x+2) <0

            ⇔−(3√x+2)4√x+8 <0

                 Vì x≥0;x≠1

                              Do đó 0<4√x+8

   Mà −(3√x+2)<0

          Vậy P<154 (đpcm)

c)Ta có:P=(3√x+8)(√x+2) 

             ⇔P=3√x+6+2(√x+2) 

             ⇔P=3(√x+2)(√x+2) +22√x+2 

              ⇔P=3+2√x+2 

Vì x≥0;x≠1⇒2√x+2 ≤1

       Do đó 

\(\dfrac{2\sqrt{X}-9}{x-5\sqrt{X}+6}-\dfrac{\sqrt{X}+3}{\sqrt{X}-2}-\dfrac{2\sqrt{X}+1}{3-\sqrt{X}}\) \(\left(X\ne2;X\ne3,X\ge0\right)\)

\(=\dfrac{2\sqrt{X}-9-\left(\sqrt{X}+3\right)\left(\sqrt{X}-3\right)+\left(2\sqrt{X}+1\right)\left(\sqrt{X}-2\right)}{\left(\sqrt{X}-2\right)\left(\sqrt{X}-3\right)}\)

\(=\dfrac{2\sqrt{X}-9-X+9+2X-4\sqrt{X}+\sqrt{X}-2}{\left(\sqrt{X}-2\right)\left(\sqrt{X}-3\right)}\)

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

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

\(C=\dfrac{\sqrt{X}+1}{\sqrt{X}-3}< 1\)

\(\Rightarrow\dfrac{\sqrt{X}+1-\sqrt{X}+3}{\sqrt{X}-3}< 0\)

\(\Rightarrow\dfrac{4}{\sqrt{X}+3}< 0\) ( VÔ LÍ)

⇒ Không có X thỏa mãn

đkxđ

\(\left\{{}\begin{matrix}x\ge0\\\sqrt{x}-3\ne0\\\sqrt{x}-2\ne0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\ge0\\x\ne9\\x\ne4\end{matrix}\right.\)

vậy \(x\ge0;x\ne4;x\ne9\)là đkxđ củaP

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

=>\(\sqrt{x-3}+1=3\)=>x-3 =4 =>x=7

ban oi ban giai thich ro o dau ban thu nhat di

a) \(\left(2\sqrt{x}-3\right)\left(2+\sqrt{x}\right)+6=0\left(ĐK:x\ge0\right)\)

\(\Leftrightarrow4\sqrt{x}+2x-6-3\sqrt{x}+6=0\)

\(\Leftrightarrow2x+\sqrt{x}=0\)

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

\(\Leftrightarrow\left[\begin{array}{nghiempt}\sqrt{x}=0\\2\sqrt{x}+1=0\left(loại\right)\end{array}\right.\)\(\Leftrightarrow x=0\)

b)\(\sqrt{x^2-9}-3\sqrt{x-3}=0\left(ĐK:x\ge3\right)\)

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

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

\(\Leftrightarrow\left[\begin{array}{nghiempt}\sqrt{x-3}=0\\\sqrt{x+3}-3=0\end{array}\right.\)\(\Leftrightarrow\left[\begin{array}{nghiempt}x=3\left(tm\right)\\x=6\left(tm\right)\end{array}\right.\)

Cho \(5\sqrt{x}7\) mk viet nham

Sua lai thanh \(5\sqrt{x}-7\)

a: \(A=\left(\dfrac{2}{\sqrt{x}-2}+\dfrac{3}{2\sqrt{x}+1}-\dfrac{5\sqrt{x}-7}{\left(2\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\right)\cdot\dfrac{5\sqrt{x}\left(\sqrt{x}-2\right)}{2\sqrt{x}+3}\)

\(=\dfrac{4\sqrt{x}+2+3\sqrt{x}-6-5\sqrt{x}+7}{\left(2\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\cdot\dfrac{5\sqrt{x}\left(\sqrt{x}-2\right)}{2\sqrt{x}+3}\)

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

b: Để A là số nguyên thì \(5\sqrt{x}⋮2\sqrt{x}+1\)

=>10 căn x+5-5 chia hết cho 2 căn x+1

=>\(2\sqrt{x}+1\in\left\{1;5\right\}\)

hay \(x\in\varnothing\)

\(B=\dfrac{x+3\sqrt{x}+2+2x-4\sqrt{x}-5\sqrt{x}-2}{x-4}=\dfrac{3x-6\sqrt{x}}{x-4}=\dfrac{3\sqrt{x}}{\sqrt{x}+2}\)

B=2/3A

=>3căn x/căn x+2=2/3*3=2

=>3căn x=2căn x+4

=>x=16

3:

a: ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\x< >9\end{matrix}\right.\)

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

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

\(=\dfrac{6}{3\left(\sqrt{x}+3\right)}=\dfrac{2}{\sqrt{x}+3}\)

b: M>1/3

=>M-1/3>0

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

=>\(\dfrac{6-\sqrt{x}-3}{3\left(\sqrt{x}+3\right)}>0\)

=>\(3-\sqrt{x}>0\)

=>\(\sqrt{x}< 3\)

=>0<=x<9

c: \(\sqrt{x}+3>=3\) với mọi x thỏa mãn ĐKXĐ

=>\(M=\dfrac{2}{\sqrt{x}+3}< =\dfrac{2}{3}\) với mọi x thỏa mãn ĐKXĐ

Dấu = xảy ra khi x=0