Answer
the solution set is\[\left\{ -\frac{2}{3},4 \right\}\].
Work Step by Step
The given equation can be written as\[\frac{3}{4}{{x}^{2}}-\frac{5}{2}x-2=0\].
Take the LCM so that this equation can be rewritten as \[3{{x}^{2}}-10x-8=0\].
Compare the given equation with the equation\[a{{x}^{2}}+bx+c=0\], where\[a=3,\ b=-10,\text{ and }c=-8\]
Now, put these values in the formula
\[\begin{align}
& x=\frac{-\left( -10 \right)\pm \sqrt{{{\left( -10 \right)}^{2}}-4\times \left( 3 \right)\times \left( -8 \right)}}{2\times 3} \\
& =\frac{-\left( -10 \right)\pm \sqrt{100+96}}{6} \\
& =\frac{10\pm \sqrt{196}}{6} \\
& =\frac{10\pm 14}{6}
\end{align}\]
Further simplifying
\[\begin{align}
& x=\frac{10\pm 14}{6} \\
& =\frac{24}{6},\frac{-4}{6} \\
& =4,-\frac{2}{3}
\end{align}\]
Hence, the solution set is\[\left\{ -\frac{2}{3},4 \right\}\].
