Answer
the solution set is\[\left\{ -\frac{3}{2},3 \right\}\].
Work Step by Step
The given equation can be written as\[\frac{{{x}^{2}}}{3}-\frac{x}{2}-\frac{3}{2}=0\].
Take the LCM so that this equation can be rewritten as \[2{{x}^{2}}-3x-9=0\].
Compare the given equation with the equation\[a{{x}^{2}}+bx+c=0\], where\[a=2,\ b=-3,\text{ and }c=-9\].
Now, put these values in the formula
\[\begin{align}
& x=\frac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\times \left( 2 \right)\times \left( -9 \right)}}{2\times 2} \\
& =\frac{3\pm \sqrt{9+72}}{4} \\
& =\frac{3\pm \sqrt{81}}{4} \\
& =\frac{3\pm 9}{4}
\end{align}\]
Further simplifying
\[\begin{align}
& x=\frac{3\pm 9}{4} \\
& =\frac{12}{4},\frac{-6}{4} \\
& =3,-\frac{3}{2}
\end{align}\]
Hence, the solution set is\[\left\{ -\frac{3}{2},3 \right\}\].
