x^2-5x+8>0 inequation

Given the inequality:
$$\left(x^{2} - 5 x\right) + 8 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} - 5 x\right) + 8 = 0$$
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 - it is the discriminant.
Because
$$a = 1$$
$$b = -5$$
$$c = 8$$
, then

D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (1) * (8) = -7

Because D<0, then the equation
has no real roots,
but complex roots is exists.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{7} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{7} i}{2}$$
$$x_{1} = \frac{5}{2} + \frac{\sqrt{7} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{7} i}{2}$$
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

$$\left(0^{2} - 0 \cdot 5\right) + 8 > 0$$

8 > 0

so the inequality is always executed