a ) \(ĐKXĐ:x\ge0;x\ne1\)

= \(\frac{x+1+\sqrt{x}}{x+1}:\left[\frac{1}{\sqrt{x}-1}-\frac{2\sqrt{x}}{\left(x+1\right)\left(\sqrt{x}-1\right)}\right]-1\)

\(=\frac{x+1+\sqrt{x}}{x+1}:\frac{x+1-2\sqrt{x}}{\left(x+1\right)\left(\sqrt{x}-1\right)}-1\)

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

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

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

B ) Ta có :

 \(Q=P-\sqrt{x}\)

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

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

Đế Q nhận giá trị nguyên thì \(1+\frac{3}{\sqrt{x}-1}\in Z\)

\(\Leftrightarrow\frac{3}{\sqrt{x}-1}\in Z\left(vì1\in Z\right)\)

\(\Leftrightarrow\sqrt{x}-1\inƯ\left(3\right)\)

Ta có bảng sau :

Vậy để biểu thức \(Q=P-\sqrt{x}\) nhận giá trị nguyên thì \(x\in\left\{16;4;0\right\}\)


 

a. ĐK \(\hept{\begin{cases}x\ge0\\x\ne2\end{cases}}\)

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

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

b. \(P\ge0\Rightarrow1-x\ge0\Rightarrow x\le1\)

Vậy với \(x\le1\)thì P có giá trị không âm

Bạn có thể diến giải phần rút gọn cho mk đc k ?

\(\left(\dfrac{x-3\sqrt{x}}{x-9}-1\right):\left(\dfrac{9-x}{x+\sqrt{x}-6}+\dfrac{\sqrt{x}-3}{\sqrt{x}-2}-\dfrac{\sqrt{x}-2}{\sqrt{x}+3}\right)\left(x\ge0;x\ne3;x\ne-3;x\ne9;x\ne4\right)\)

\(=\left(\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-1\right):\left(\dfrac{9-x}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}+\dfrac{\sqrt{x}-3}{\sqrt{x}-2}-\dfrac{\sqrt{x}-2}{\sqrt{x}+3}\right)\\ =\dfrac{\sqrt{x}-\sqrt{x}-3}{\sqrt{x}+3}:\dfrac{9-x+\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\\ =\dfrac{-3}{\sqrt{x}+3}:\dfrac{9-x+x-9-x+4\sqrt{x}-4}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\\ =\dfrac{-3}{\sqrt{x}+3}\cdot\dfrac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}{-\left(\sqrt{x}-2\right)^2}\\ =\dfrac{3}{\sqrt{x}-2}\)

Tick hộ nha 😘

a)\(x-4\ne0;x\ge0\)

<=>\(x\ne4;x\ge0\)

b)\(B=\left(\frac{1}{\sqrt{x}+2}-\frac{2}{x+4\sqrt{x}+4}\right):\left(\frac{2}{x-4}-\frac{1}{\sqrt{x}-2}\right)\)

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

=\(\left(\frac{\sqrt{x}+2}{\left(\sqrt{x}+2\right)^2}-\frac{2}{\left(\sqrt{x}+2\right)^2}\right):\left(\frac{2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\frac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right)\)

=\(\frac{\sqrt{x}}{\left(\sqrt{x}+2\right)^2}:\frac{\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

=\(\frac{\sqrt{x}}{\left(\sqrt{x}+2\right)^2}.\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\sqrt{x}}\)

=\(\frac{\sqrt{x}-2}{\sqrt{x}+2}\)