Question

giải pt vô tỉ:

\(\sqrt{2x^2-9x+4}+3\sqrt{2x-1}=\sqrt{2x^2+21x-11}\)

Answer 1

\(\Leftrightarrow\sqrt{\left(2x-1\right)\left(x-4\right)}+3\sqrt{2x-1}=\sqrt{\left(2x-1\right)\left(x+11\right)}\)

ĐK \(x\ge\dfrac{1}{2}\)

\(\Leftrightarrow\sqrt{\left(2x-1\right)\left(x-4\right)}+3\sqrt{2x-1}-\sqrt{\left(2x-1\right)\left(x+11\right)}=0\)

\(\Leftrightarrow\sqrt{2x-1}\left(\sqrt{x-4}+3-\sqrt{x+11}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}\left(nh\right)\\\sqrt{x-4}+3=\sqrt{x+11}\end{matrix}\right.\)

\(\Leftrightarrow\left(\sqrt{x-4}+3\right)^2=x+11\)

\(\Leftrightarrow x+5+6\sqrt{x-4}=x+11\)

\(\Leftrightarrow x+5+6\sqrt{x-4}=x+11\)

\(\Leftrightarrow\sqrt{x-4}=1\)

\(\Leftrightarrow x=5\left(nh\right)\)

vậy \(S=\left\{\dfrac{1}{2};5\right\}\)