2*x^2-x+2<0 inequation

Given the inequality:
$$2 x^{2} - x + 2 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 x^{2} - x + 2 = 0$$
Solve:
This equation is of the form
$$a*x^2 + b*x + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = -1$$
$$c = 2$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 2 \cdot 4 \cdot 2 + \left(-1\right)^{2} = -15$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{1}{4} + \frac{\sqrt{15} i}{4}$$
Simplify
$$x_{2} = \frac{1}{4} - \frac{\sqrt{15} i}{4}$$
Simplify
$$x_{1} = \frac{1}{4} + \frac{\sqrt{15} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{15} i}{4}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$2 \cdot 0^{2} - 0 + 2 < 0$$

2 < 0

but

2 > 0

so the inequality has no solutions