The given equation is $2 x^{2}+x-4=0$

Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=1$ and $c=-4$

$\therefore$ Discriminant, $D=b^{2}-4 a c=(1)^{2}-4 \times 2 \times(-4)=1+32=33>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{33}$

$\begin{array}{l}
\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-1+\sqrt{33}}{2 \times 2}=\frac{-1+\sqrt{33}}{4} \\
\beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-1-\sqrt{33}}{2 \times 2}=\frac{-1-\sqrt{33}}{4}
\end{array}$