Solve 3x² − 2√6x + 2 = 0 by Factorization
Question:
\[ 3x^2-2\sqrt6\,x+2=0 \]Solution
Given:
\[ 3x^2-2\sqrt6\,x+2=0 \]Product of the coefficient of \(x^2\) and the constant term:
\[ 3\times 2=6 \]We split the middle term \(-2\sqrt6\,x\) as \(-\sqrt6\,x-\sqrt6\,x\):
\[ 3x^2-\sqrt6\,x-\sqrt6\,x+2=0 \]Taking common factors:
\[ \sqrt3\,x(\sqrt3\,x-\sqrt2)-\sqrt2(\sqrt3\,x-\sqrt2)=0 \] \[ (\sqrt3\,x-\sqrt2)(\sqrt3\,x-\sqrt2)=0 \] \[ (\sqrt3\,x-\sqrt2)^2=0 \]Therefore,
\[ \sqrt3\,x-\sqrt2=0 \] \[ x=\frac{\sqrt2}{\sqrt3} \] \[ x=\frac{\sqrt6}{3} \]Final Answer
\[ \boxed{x=\frac{\sqrt6}{3}} \]Repeated Root
\[ \boxed{\left(x-\frac{\sqrt6}{3}\right)^2=0} \]