Answer
$x=-1+\dfrac{\sqrt{22}}{2}$ and $x=-1-\dfrac{\sqrt{22}}{2}$ are not part of the domain of the given expression.
Work Step by Step
$\dfrac{3}{2x^{2}+4x-9}$
The numbers that are not included in the domain of this expression are the values of $x$ for which the denominator is equal to $0$.
Set the denominator equal to $0$:
$2x^{2}+4x-9=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
In this case, $a=2$, $b=4$ and $c=-9$
Substitute the known values into the formula and evaluate:
$x=\dfrac{-4\pm\sqrt{4^{2}-4(2)(-9)}}{2(2)}=\dfrac{-4\pm\sqrt{16+72}}{4}=...$
$...=\dfrac{-4\pm\sqrt{88}}{4}=\dfrac{-4\pm2\sqrt{22}}{4}=-1\pm\dfrac{\sqrt{22}}{2}$
$x=-1+\dfrac{\sqrt{22}}{2}$ and $x=-1-\dfrac{\sqrt{22}}{2}$ are not part of the domain of the given expression.
