Question

Cho biểu thức:

P = \(\dfrac{3}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-5}{x-1}\) Với x > 0 ; x khác 1

a) Rút gọn P

b) Tính P khi x = \(24-16\sqrt{2}\)

Answer 1

\(a.P=\dfrac{3}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-5}{x-1}=\dfrac{3\sqrt{x}-3-\sqrt{x}-1-\sqrt{x}+5}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{1}{\sqrt{x}-1}\left(x>0;x\ne1\right)\)

\(b.x=24-16\sqrt{2}=16-2.2\sqrt{2}.4+8=\left(4-2\sqrt{2}\right)^2\left(TM\right)\)

\(\Rightarrow\sqrt{x}=4-2\sqrt{2}\)

Khi đó :\(P=\dfrac{1}{4-2\sqrt{2}-1}=\dfrac{1}{3-2\sqrt{2}}=\dfrac{1}{2-2\sqrt{2}+1}=\dfrac{1}{\left(\sqrt{2}-1\right)^2}\)