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\(1,P=\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{\sqrt{x}+1}{x-1}\)
\(=\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{\left(x+\sqrt{x}+1\right)}-\frac{\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{\left(x+\sqrt{x}+1\right)}-\frac{1}{\sqrt{x}-1}\)
\(=\frac{x+2}{x\sqrt{x}-1}+\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(x+\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\frac{x+2+x-1-x-\sqrt{x}-1}{x\sqrt{x}-1}\)
\(=\frac{x-\sqrt{x}}{x\sqrt{x}-1}\)
\(=\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\frac{\sqrt{x}}{x+\sqrt{x}+1}\)
Câu 1 bạn ghi lại đề, tử số cuối cùng bị thiếu
2/ \(a^2-4a+1=0\Rightarrow a^2+1=4a\Rightarrow a+\frac{1}{a}=4\)
\(\Rightarrow a^2+\frac{1}{a^2}+2=16\Rightarrow a^2+\frac{1}{a^2}=14\)
\(P=\frac{a^4+a^2+1}{a^2}=a^2+\frac{1}{a^2}+1=14+1=15\)
ĐKXĐ: \(x\ge0;x\ne1\)
\(Q=\frac{\left(\sqrt{x}+1\right)^2}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}-1\right)^2}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{4\left(\sqrt{x}+1\right)}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{x+2\sqrt{x}+1-x+2\sqrt{x}-1-4\sqrt{x}-4}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\frac{-4}{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{-2}{x-1}\)
Để \(Q>1\Rightarrow\frac{-2}{x-1}>1\Rightarrow\frac{x+1}{x-1}< 0\Rightarrow x-1< 0\Rightarrow x< 1\)
Vậy \(0\le x< 1\)
a: ĐKXĐ: x>0; x<>1
b: \(B=\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)^2\cdot\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}+1\right)\left(x-1\right)}{\sqrt{x}}\)
\(=\dfrac{x+\sqrt{x}-2-x+\sqrt{x}+2}{\sqrt{x}}=2\)
1, a, ĐKXĐ: x > 0
\(\Rightarrow P=\frac{\sqrt{x}\left(x\sqrt{x}+1\right)}{x-\sqrt{x}+1}-\left(2\sqrt{x}+1\right)+1\)
\(\Rightarrow P=\frac{\sqrt{x}\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{x-\sqrt{x}+1}-2\sqrt{x}-1+1\)
\(\Rightarrow P=\sqrt{x}\left(\sqrt{x}+1\right)-2\sqrt{x}\)
\(\Rightarrow P=x+\sqrt{x}-2\sqrt{x}\)
\(\Rightarrow P=x-\sqrt{x}\)
b, Thay x=100 vào biểu thức P, ta có:
P= 100 - \(\sqrt{100}\)
\(\Rightarrow P=100-10=90\)
Vậy với x=100 thì P=90
c, Ta có: P= \(x-\sqrt{x}=\left(\sqrt{x}-\frac{1}{2}\right)^2-\frac{1}{4}\ge-\frac{1}{4}\)
Dấu "=" xảy ra khi...
2, a, ĐKXĐ: x \(\ge\) 0, x \(\ne\) 1
\(\Rightarrow A=\left(\frac{x+3\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}+2}\right):\frac{1}{x-1}\)
\(\Rightarrow A=\left(\frac{x+3\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\frac{\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}-\frac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\right).\frac{x-1}{1}\)
\(\Rightarrow A=\left(\frac{x+3\sqrt{x}-1-\sqrt{x}-2-\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\right).\frac{x-1}{1}\)
\(\Rightarrow\)A= \(\frac{x+\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}.\frac{x-1}{1}\)= x-1
b, Để \(\frac{1}{A}\)là số tự nhiên (x \(\ge0\), \(x\ne1\))
\(\Rightarrow x-1=1\)
\(\Rightarrow x=2\)
Vậy x=2 thì \(\frac{1}{A}\) là số tự nhiên.
Ta có: