Solve the Following Quadratic Equation by Factorization
Question:
\[ \sqrt{3}x^2-2\sqrt{2}x-2\sqrt{3}=0 \]Solution
Given:
\[ \sqrt{3}x^2-2\sqrt{2}x-2\sqrt{3}=0 \]Product of the coefficient of \(x^2\) and the constant term:
\[ (\sqrt{3})(-2\sqrt{3})=-6 \]We split the middle term \(-2\sqrt{2}x\) as \(-3\sqrt{2}x+\sqrt{2}x\):
\[ \sqrt{3}x^2-3\sqrt{2}x+\sqrt{2}x-2\sqrt{3}=0 \]Taking common factors:
\[ x(\sqrt{3}x-3\sqrt{2}) +\sqrt{2}(\,x-\sqrt{6}\,) =0 \] \[ x(\sqrt{3}x-3\sqrt{2}) +\frac{\sqrt{2}}{\sqrt{3}}(\sqrt{3}x-3\sqrt{2}) =0 \] \[ (\sqrt{3}x-3\sqrt{2}) \left(x+\frac{\sqrt{6}}{3}\right)=0 \]Therefore,
\[ \sqrt{3}x-3\sqrt{2}=0 \quad \text{or} \quad x+\frac{\sqrt{6}}{3}=0 \] \[ x=\frac{3\sqrt{2}}{\sqrt{3}} =\sqrt{6} \] \[ x=-\frac{\sqrt{6}}{3} \]