-7x²-21x<0 inequation

Given the inequality:
$$- 7 x^{2} - 21 x < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 7 x^{2} - 21 x = 0$$
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
$$a = -7$$
$$b = -21$$
$$c = 0$$
, then

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

(-21)^2 - 4 * (-7) * (0) = 441

Because D > 0, then the equation has two roots.

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

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

or
$$x_{1} = -3$$
$$x_{2} = 0$$
$$x_{1} = -3$$
$$x_{2} = 0$$
$$x_{1} = -3$$
$$x_{2} = 0$$
This roots
$$x_{1} = -3$$
$$x_{2} = 0$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-3 + - \frac{1}{10}$$
=
$$- \frac{31}{10}$$
substitute to the expression
$$- 7 x^{2} - 21 x < 0$$
$$- 7 \left(- \frac{31}{10}\right)^{2} - \frac{\left(-31\right) 21}{10} < 0$$

-217     
----- < 0
 100     

one of the solutions of our inequality is:
$$x < -3$$

 _____           _____          
      \         /
-------ο-------ο-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -3$$
$$x > 0$$