Question

Cho biểu thức \(P=\left(\dfrac{x\sqrt{x}-1}{x-\sqrt{x}}-\dfrac{x\sqrt{x}+1}{x+\sqrt{x}}\right)\div[\dfrac{2\left(x-2\sqrt{x}+1\right)}{x-1}\)

a. rút gọn P

b. Tìm x để P<0

c. Tìm x nguyên để Pcos giá trị nguyên

Answer 1

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

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

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

b Để P<0 thì \(\sqrt{x}-1< 0\)

=>0<x<1

c: Để P là số nguyên thì \(\sqrt{x}-1+2⋮\sqrt{x}-1\)

\(\Leftrightarrow\sqrt{x}-1\in\left\{1;-1;2;-2\right\}\)

hay \(x\in\left\{4;9\right\}\)