Question

Cho biểu thức \(P=\left(\frac{1}{\sqrt{x}-\sqrt{x-1}}-\frac{x-3}{\sqrt{x-1}-\sqrt{2}}\right)\left(\frac{2}{\sqrt{2}-\sqrt{x}}-\frac{\sqrt{x}+\sqrt{2}}{\sqrt{2x}-x}\right)\)

a) Rút gọn P

b) Tính giá trị của P với \(x=3-2\sqrt{2}\)

Answer 1

ĐKXĐ: ....

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

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

\(x=3-2\sqrt{2}=\left(\sqrt{2}-1\right)^2\Rightarrow\sqrt{x}=\sqrt{2}-1\)

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