Question

Cho \(A=\left(\frac{3}{x-1}+\frac{1}{\sqrt{x}+1}\right):\frac{1}{\sqrt{x}+1}\)

a) Tìm ĐKXĐ và rút gọn

b)Tính A khi x=\(\frac{4}{9}\)

c)Tính giá trị của x để A=\(\frac{5}{4}\)

Answer 1

a) \(ĐKXĐ:\hept{\begin{cases}x\ge0\\x-1\ne0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)

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

       \(=\left[\frac{3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}+\frac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right].\left(\sqrt{x}+1\right)\)

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

b) Ta có: \(x=\frac{4}{9}\)thỏa mãn ĐKXĐ

  \(\Rightarrow\)Thay \(x=\frac{4}{9}\)vào biểu thức A ta có:

\(A=\frac{\sqrt{\frac{4}{9}}+2}{\sqrt{\frac{4}{9}}-1}=\frac{\frac{2}{3}+2}{\frac{2}{3}-1}=\frac{\frac{8}{3}}{-\frac{1}{3}}=-8\)

c) Ta có: \(A=\frac{5}{4}\)\(\Leftrightarrow\frac{\sqrt{x}+2}{\sqrt{x}-1}=\frac{5}{4}\)

\(\Leftrightarrow4\left(\sqrt{x}+2\right)=5\left(\sqrt{x}-1\right)\)\(\Leftrightarrow4\sqrt{x}+8=5\sqrt{x}-5\)

\(\Leftrightarrow\sqrt{x}=13\)\(\Leftrightarrow x=169\)( thỏa mãn ĐKXĐ )

 Vậy \(x=169\)

Answer 2

\(a,ĐKXĐ:x\ne1,x>0\)

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

\(A=\frac{3+\sqrt{x}-1}{x-1}.\frac{\sqrt{x}+1}{1}\)

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

với \(x=\frac{4}{9}\)

\(< =>A=\frac{2+\sqrt{\frac{4}{9}}}{\sqrt{\frac{4}{9}}-1}\)

\(A=\frac{2+\frac{2}{3}}{\frac{2}{3}-1}=\frac{\frac{8}{3}}{\frac{-1}{3}}=-8\)

\(c,\frac{5}{4}=\frac{2+\sqrt{x}}{\sqrt{x}-1}\)

\(5\sqrt{x}-5=8+4\sqrt{x}\)

\(\sqrt{x}=13< =>x=169\)

Answer 3

a, \(A=\left(\frac{3}{x-1}+\frac{1}{\sqrt{x}+1}\right):\frac{1}{\sqrt{x}+1}\)ĐK : \(x\ne1;x\ge0\)

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

b, Thay \(x=\frac{4}{9}\Rightarrow\sqrt{x}=\sqrt{\frac{4}{9}}=\frac{2}{3}\)vào biểu thức A ta được 

\(A=\frac{\frac{2}{3}+2}{\frac{2}{3}-1}=\frac{\frac{8}{3}}{-\frac{1}{3}}=\frac{8}{3}.\frac{-3}{1}=-8\)

Vậy với x = 4/9 thì A = -8

c, Ta có : \(A=\frac{5}{4}\Rightarrow\frac{\sqrt{x}+2}{\sqrt{x}-1}=\frac{5}{4}\Rightarrow4\sqrt{x}+8=5\sqrt{x}-5\)

\(\Leftrightarrow\sqrt{x}=13\Leftrightarrow x=169\)

Vậy với A = 5/4 thì x = 169