Question

Giải phương trình \(\sqrt{4x^2+5x+1}-2\sqrt{x^2-1+1}=3-9x\)

Answer 1

ĐKXĐ: \(\left[{}\begin{matrix}x\le-1\\x\ge-\frac{1}{4}\end{matrix}\right.\)

\(\sqrt{4x^2+5x+1}-\sqrt{4x^2-4x+4}+9x-3=0\)

Đặt \(\left\{{}\begin{matrix}a=\sqrt{4x^2+5x+1}\ge0\\b=\sqrt{4x^2-4x+4}>0\end{matrix}\right.\) \(\Rightarrow a^2-b^2=9x-3\)

Phương trình trở thành:

\(a-b+a^2-b^2=0\)

\(\Leftrightarrow a-b+\left(a-b\right)\left(a+b\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(1+a+b\right)=0\)

\(\Leftrightarrow a=b\) (do \(\left\{{}\begin{matrix}a\ge0\\b>0\end{matrix}\right.\Rightarrow1+a+b>0\))

\(\Rightarrow\sqrt{4x^2+5x+1}=\sqrt{4x^2-4x+4}\)

\(\Leftrightarrow9x=3\)

\(\Rightarrow x=\frac{1}{3}\)