Question

Cho P=\(\frac{1}{\sqrt{x+2\sqrt{x-1}}}+\frac{1}{\sqrt{x-2\sqrt{x-1}}}\)

a) Rút gọn P.

b) Tìm x để P=1/4.

c) Tính P biết \(x=5+2\sqrt{3}\).

Answer 1

a)ĐK: x>1

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

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

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

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

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

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