Question

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

a) Rút gọn Q

b) Tính M khi x = 4 - \(2\sqrt{3}\)

c) Tìm x để M = \(\frac{\sqrt{x}}{6}\)

d) Tìm GTLN của M

Answer 1

a,

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

\(M=\left(\frac{2\sqrt{x}+6+\sqrt{x}-3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right):\frac{\sqrt{x}+1}{\sqrt{x}-3}\)

\(M=\frac{3\sqrt{x}+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}.\frac{\sqrt{x}-3}{\sqrt{x}+1}\)

\(M=\frac{3\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}.\frac{\sqrt{x}-3}{\sqrt{x}+1}\)

\(M=\frac{3}{\sqrt{x}+1}\)

b, khi x = \(4-2\sqrt{3}=\left(\sqrt{3}-1\right)^2\)

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