Question

Đề bài: \(P=\left(\dfrac{1}{1-\sqrt{a}}-\dfrac{1}{1+\sqrt{a}}\right).\left(1-\dfrac{1}{\sqrt{a}}\right)\left(a>0;a\ne1\right)\)

a) Rút gọn

b) Tìm a để \(P=-\dfrac{1}{2}\)

Answer 1

a) Ta có: \(P=\left(\dfrac{1}{1-\sqrt{a}}-\dfrac{1}{1+\sqrt{a}}\right)\cdot\left(1-\dfrac{1}{\sqrt{a}}\right)\)

\(=\dfrac{1+\sqrt{a}-1+\sqrt{a}}{-\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\cdot\dfrac{\sqrt{a}-1}{\sqrt{2}}\)

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

b) Ta có: \(P=\dfrac{-1}{2}\)

nên \(\dfrac{2}{\sqrt{a}+1}=\dfrac{1}{2}\)

\(\Leftrightarrow\sqrt{a}+1=4\)

\(\Leftrightarrow a=9\)(thỏa ĐK)