-7x^2+5x-2<0 inequation

Given the inequality:
$$\left(- 7 x^{2} + 5 x\right) - 2 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 7 x^{2} + 5 x\right) - 2 = 0$$
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 - it is the discriminant.
Because
$$a = -7$$
$$b = 5$$
$$c = -2$$
, then

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

(5)^2 - 4 * (-7) * (-2) = -31

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
$$x_{1} = \frac{5}{14} - \frac{\sqrt{31} i}{14}$$
$$x_{2} = \frac{5}{14} + \frac{\sqrt{31} i}{14}$$
$$x_{1} = \frac{5}{14} - \frac{\sqrt{31} i}{14}$$
$$x_{2} = \frac{5}{14} + \frac{\sqrt{31} i}{14}$$
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

$$-2 + \left(- 7 \cdot 0^{2} + 0 \cdot 5\right) < 0$$

-2 < 0

so the inequality is always executed