Question

Cho biểu thức :

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

a, Tìm điều kiện của x để P xác định

b, Rút gọn P

c, Tìm x để P = \(\frac{1}{4}\)

Các bạn ơi ! giúp mk với đi

Answer 1

xin lỗi, mình làm nhầm câu b/
b/ P\(=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{x-1-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{3}=\frac{\sqrt{x}-2}{3\sqrt{x}}\)
Vậy P \(=\frac{\sqrt{x}-2}{3\sqrt{x}}\) khi \(x\ne1;x\ne4;x>0\)
c/Để P =1/4 thì \(\frac{\sqrt{x}-2}{3\sqrt{x}}=\frac{1}{4}\Leftrightarrow4\sqrt{x}-8=3\sqrt{x}\Leftrightarrow4\sqrt{x}-3\sqrt{x}=8\Leftrightarrow\sqrt{x}=8\Leftrightarrow x=64\)
Chúc bạn học tốt!!!

Answer 2

a/ ĐKXĐ:\(x>0;x\ne1;x\ne4\)
b/Ta có:\(P=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)
\(=\frac{\sqrt{x}-\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{x-1-x+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{1}\)
\(=\frac{\sqrt{x}-2}{\sqrt{x}}\)
Vậy P \(=\frac{\sqrt{x}-2}{\sqrt{x}}\) khi \(x>0;x\ne1;x\ne4\)
c/Để P=\(\frac{1}{4}\)\(\Leftrightarrow\frac{\sqrt{x}-2}{\sqrt{x}}=\frac{1}{4}\Leftrightarrow4\left(\sqrt{x}-2\right)=\sqrt{x}\Leftrightarrow4\sqrt{x}-8=\sqrt{x}\Leftrightarrow4\sqrt{x}-\sqrt{x}=8\Leftrightarrow3\sqrt{x}=8\Leftrightarrow\sqrt{x}=\frac{8}{3}\Leftrightarrow x=\frac{64}{9}\)(thỏa mãn)
Vậy x=\(\frac{64}{9}\) thì P=\(\frac{1}{4}\)
Chúc bạn học tốt!