Question

Câu 1: Tính giá trị biểu thức: \(A=\frac{\sqrt{x}+1}{\sqrt{x}-1}\) khi x = 9

Câu 2: Cho biểu thức P = \(\left(\frac{x-2}{x+2\sqrt{x}}+\frac{1}{\sqrt{x}+2}\right).\frac{\sqrt{x}+1}{\sqrt{x}-1}\) với \(x>0,x\ne1\)

a) Chứng minh \(P=\frac{\sqrt{x}+1}{\sqrt{x}}\)

b) Tìm giá trị của x để \(2P=2\sqrt{x}+5\)

Answer 1

1/ ĐKXĐ : \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)

Thay \(x=9\) vào biểu thức A ta có :

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

Vậy...

2/ ĐKXĐ : \(\left\{{}\begin{matrix}x>0\\x\ne1\end{matrix}\right.\)

Ta có :

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

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

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

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

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

Vậy....

b/ Ta có :

\(2P=2\sqrt{x}+5\)

\(\Leftrightarrow\frac{2\left(\sqrt{x}+1\right)}{\sqrt{x}}=2\sqrt{x}+5\)

\(\Leftrightarrow2\sqrt{x}+2=2x+5\sqrt{x}\)

\(\Leftrightarrow2x+3\sqrt{x}-2=0\)

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

\(\Leftrightarrow2\sqrt{x}-1=0\)

\(\Leftrightarrow x=\frac{1}{4}\)

Vậy..

Answer 2

Câu 1 :

\(A=\frac{\sqrt{x}+1}{\sqrt{x-1}}\) khi x = 9

tại x = 9 thay vào A ta được : \(\frac{\sqrt{9}+1}{\sqrt{9}-1}\) = \(\frac{3+1}{3-1}=\frac{4}{2}=2\)

Câu 2 :

a, Ta có : P = \(\left(\frac{x-2}{x+2\sqrt{x}}+\frac{1}{\sqrt{x}+2}\right).\frac{\sqrt{x}+1}{\sqrt{x}-1}\)

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

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

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

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

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

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

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

\(=\frac{\sqrt{x}+1}{\sqrt{x}}\) => đpcm