Question

giải phương trình: \(2\left(x+1\right)\sqrt{x+1}=\left(\sqrt{x+1}+\sqrt{1-x}\right)\left(2-\sqrt{1-x^2}\right)\)

Answer 1

ĐKXĐ...

Đặt \(\left\{{}\begin{matrix}\sqrt{1+x}=a\\\sqrt{1-x}=b\end{matrix}\right.\) \(\Rightarrow a^2+b^2=2\)

\(2a^3=\left(a+b\right)\left(a^2+b^2-ab\right)\)

\(\Leftrightarrow2a^3=\left(a+b\right)^3\)

\(\Leftrightarrow a\sqrt[3]{2}=a+b\)

\(\Leftrightarrow a\left(\sqrt[3]{2}-1\right)=b\)

\(\Rightarrow\sqrt{1+x}\left(\sqrt[3]{2}-1\right)=\sqrt{1-x}\)

\(\Leftrightarrow\left(1+x\right)\left(\sqrt[3]{2}-1\right)^2=1-x\)

\(\Rightarrow x=\frac{1-\left(\sqrt[3]{2}-1\right)^2}{\left(\sqrt[3]{2}-1\right)^2+1}\)