Question

Tính giá trị của biểu thức:

\(C=\frac{1+2x}{1+\sqrt{1+2x}}+\frac{1-2x}{1-\sqrt{1-2x}}\) khi \(x=\frac{\sqrt{3}}{4}\)

Answer 1

\(C=\frac{2+4x}{2+\sqrt{4+8x}}+\frac{2-4x}{2-\sqrt{4-8x}}\)

Với \(x=\frac{\sqrt{3}}{4}\)

\(\Rightarrow C=\frac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2-\sqrt{3}}{2-\sqrt{4-2\sqrt{3}}}=\frac{2+\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{2-\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)

\(=\frac{2+\sqrt{3}}{3+\sqrt{3}}+\frac{2-\sqrt{3}}{3-\sqrt{3}}=\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{6}\)

\(=\frac{3+\sqrt{3}+3-\sqrt{3}}{6}=1\)