Question

Giải phương trình

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

Answer 1

ĐKXĐ: \(0\le x\le1\)

Đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\ge0\\\sqrt{1-x}=b\ge0\end{matrix}\right.\) ta được hệ:

\(\left\{{}\begin{matrix}a^2+b^2=1\\\frac{2a^3}{a+b}+ab=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2a^3+ab\left(a+b\right)=a+b\\a^2+b^2=1\end{matrix}\right.\)

\(\Rightarrow2a^3+a^2b+ab^2=\left(a+b\right)\left(a^2+b^2\right)\)

\(\Rightarrow2a^3+a^2b+ab^2=a^3+ab^2+ab^2+b^3\)

\(\Leftrightarrow a^3=b^3\)

\(\Leftrightarrow a=b\Leftrightarrow\sqrt{x}=\sqrt{1-x}\)

\(\Leftrightarrow x=1-x\Leftrightarrow x=\frac{1}{2}\)