Question

Giải các phương trình:

a) \(\left(3x-1\right)\left(3x+1\right)=x\left(1+8\sqrt{x+1}\right)\)

b) \(5x^2-5x\sqrt{x^2+x+4}+2x+5=0\)

c) \(9x^2+8x+9=9\left(x+1\right)\sqrt{2x^2+1}\)

d) \(5x^2+2x+2=5x\sqrt{x^2+x+1}\)

e) \(5x^2+20x-12=5\left(x-2\right)\sqrt{3x^2+x}\)

Answer 1

a/ ĐXĐK: ...

\(\Leftrightarrow9x^2-1-x-8x\sqrt{x+1}=0\)

\(\Leftrightarrow x^2-x-1+8x\left(x-\sqrt{x+1}\right)=0\)

\(\Leftrightarrow x^2-x-1+\frac{8x\left(x^2-x-1\right)}{x+\sqrt{x+1}}=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2-x-1=0\Rightarrow x=...\\\frac{-8x}{x+\sqrt{x+1}}=1\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow-8x=x+\sqrt{x+1}\)

\(\Leftrightarrow-9x=\sqrt{x+1}\) (\(x\le0\))

\(\Leftrightarrow81x^2-x-1=0\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{1-5\sqrt{13}}{162}\\x=\frac{1+5\sqrt{13}}{162}>0\left(l\right)\end{matrix}\right.\)

Answer 2

d/

\(\Leftrightarrow3x^2+2\left(x^2+x+1\right)-5x\sqrt{x^2+x+1}=0\)

Đặt \(\sqrt{x^2+x+1}=a\)

\(\Leftrightarrow3x^2-5ax+2a^2=0\)

\(\Leftrightarrow\left(x-a\right)\left(3x-2a\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=a\\3x=2a\end{matrix}\right.\) (\(x\ge0\))

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+x+1}=x\\2\sqrt{x^2+x+1}=3x\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2+x+1=x^2\\2\left(x^2+x+1\right)=9x^2\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-1\left(l\right)\\7x^2-2x-2=0\end{matrix}\right.\) \(\Rightarrow x=\frac{1+\sqrt{15}}{7}\)