Question

Tính và rút gọn:

C=\(\dfrac{2}{\sqrt{5}+1}+\sqrt{\dfrac{2}{3-\sqrt{5}}}\)

Giai các pt sau:

\(\sqrt{3x+1}-\sqrt{x-1}\)=2

b, \(\sqrt{x-5}\)=x-7

Answer 1

C\(=\dfrac{2}{\sqrt{5}+1}+\sqrt{\dfrac{2}{3-\sqrt{5}}}\)

\(=\dfrac{2\left(\sqrt{5}-1\right)}{4}+\sqrt{\dfrac{2\left(3+\sqrt{5}\right)}{4}}\)

\(=\dfrac{\sqrt{5}-1}{2}+\sqrt{\dfrac{3+\sqrt{5}}{2}}\)

\(=\dfrac{\sqrt{5}-1}{2}+\dfrac{\sqrt{3+\sqrt{5}}}{\sqrt{2}}\)

\(=\dfrac{\sqrt{5}-1}{2}+\dfrac{\sqrt{\left(3+\sqrt{5}\right).2}}{2}\)

\(=\dfrac{\sqrt{5}-1}{2}+\dfrac{\sqrt{6+2\sqrt{5}}}{2}\)

\(=\dfrac{\sqrt{5}-1}{2}+\dfrac{\sqrt{\left(1+\sqrt{5}\right)^2}}{2}\)

\(=\dfrac{\sqrt{5}-1}{2}+\dfrac{1+\sqrt{5}}{2}\)

\(=\dfrac{\sqrt{5}-1+1+\sqrt{5}}{2}\)

\(=\dfrac{2\sqrt{5}}{2}\)

\(=\sqrt{5}\)