Question

giải phương trình:

\(1+\dfrac{2}{3}\sqrt{x-x^2}=\sqrt{x}+\sqrt{1-x}\)

Answer 1

\(1+\dfrac{2}{3}\sqrt{x-x^2}=\sqrt{x}+\sqrt{1-x}\)(ĐKXĐ:tự tìm)(1)

Đặt \(\sqrt{x}+\sqrt{1-x}=t\left(0< t< 2\right)\)

\(\Rightarrow t^2=x+1-x+2\sqrt{x\left(1-x\right)}\)

\(t^2-1=2\sqrt{x-x^2}\)

(1)\(\Leftrightarrow1+\dfrac{t^2-1}{3}=t\)

\(\Leftrightarrow3+t^2-1=3t\)

\(\Leftrightarrow t^2-3t+2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}t=1\\t=2\left(loai\right)\end{matrix}\right.\)

Với t=1\(\Rightarrow x=0\) hoặc x=1