Question

Giải phương trình: \(5+x+2\sqrt{\left(4-x\right)\left(2x-2\right)}=4\left(\sqrt{4-x}+\sqrt{2x-2}\right)\)

Answer 1

\(5+x+2\sqrt{\left(4-x\right)\left(2x-2\right)}=4\left(\sqrt{4-x}+\sqrt{2x-2}\right)\)

ĐK:\(1\le x\le 4\)

Đặt \(\left\{{}\begin{matrix}\sqrt{4-x}=a\\\sqrt{2x-2}=b\end{matrix}\right.\)\(\left(a,b\ge0\right)\) thì ta có:

\(a^2+b^2+3+2ab=4\left(a+b\right)\)

\(\Leftrightarrow\left(a+b\right)^2-4\left(a+b\right)=-3\)

\(\Leftrightarrow\left(a+b\right)\left(a+b-4\right)=-3\)

\(\Rightarrow\left[{}\begin{matrix}b=1-a\\b=3-a\end{matrix}\right.\)\(\Rightarrow\left[{}\begin{matrix}\sqrt{2x-2}=1-\sqrt{4-x}\\\sqrt{2x-2}=3-\sqrt{4-x}\end{matrix}\right.\)\(\Rightarrow x=3\)

Answer 2

Đặt \(\left[{}\begin{matrix}u=\sqrt{4-x}\ge0\\v=\sqrt{2x-2}\ge0\end{matrix}\right.\)khi đó pt đã cho trở thành :

9-u² +2uv=4(u+v)