Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

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

$\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-1+\sqrt{33}}{2 \times 2}=\frac{-1+\sqrt{33}}{4}$

$\beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-1-\sqrt{33}}{2 \times 2}=\frac{-1-\sqrt{33}}{4}$

Hence, $\frac{-1+\sqrt{33}}{4}$ and $\frac{-1-\sqrt{33}}{4}$ are the roots of the given equation.