Question

Chứng minh đẳng thức

\(\dfrac{\sqrt{a}-2}{a+2\sqrt{a}}+\dfrac{8}{a-4}=\dfrac{\sqrt{a}+2}{a-2\sqrt{a}}\) (với a > 0 ; a \(\ne\)4)

Answer 1

\(\dfrac{\sqrt{a}-2}{a+2\sqrt{a}}+\dfrac{8}{a-4}\)

\(=\dfrac{\sqrt{a}-2}{\sqrt{a}\left(\sqrt{a}+2\right)}+\dfrac{8}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\)

\(=\dfrac{\left(\sqrt{a}-2\right)^2+8\sqrt{a}}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)\cdot\sqrt{a}}\)

\(=\dfrac{\left(\sqrt{a}+2\right)^2}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)\cdot\sqrt{a}}=\dfrac{\sqrt{a}+2}{\sqrt{a}\left(\sqrt{a}-2\right)}\)

\(=\dfrac{\sqrt{a}+2}{a-2\sqrt{a}}\)