Question

Cho biểu thức:

\(B=\left(\frac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right)\left(\frac{1+a\sqrt{a}}{1+\sqrt{a}}-\sqrt{a}\right)\)

a) tìm đkxđ và rút gọn B

b) Tìm a để \(B< 7-4\sqrt{3}\)

Answer 1

ĐKXĐ: \(a\ge0;a\ne1\)

\(B=\left(\frac{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}{1-\sqrt{a}}+\sqrt{a}\right)\left(\frac{\left(1+\sqrt{a}\right)\left(a-\sqrt{a}+1\right)}{1+\sqrt{a}}-\sqrt{a}\right)\)

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

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

\(=\left(a-1\right)^2\)

Để \(B< 7-4\sqrt{3}\Rightarrow\left(a-1\right)^2< 7-4\sqrt{3}\)

\(\Rightarrow\left(a-1\right)^2< \left(2-\sqrt{3}\right)^2\Rightarrow-2+\sqrt{3}< a-1< 2-\sqrt{3}\)

\(\Rightarrow-1+\sqrt{3}< a< 3-\sqrt{3}\)

Kết hợp ĐKXĐ \(\Rightarrow\left\{{}\begin{matrix}\sqrt{3}-1< a< 3-\sqrt{3}\\a\ne1\end{matrix}\right.\)