x^2–4x<0 inequation

Given the inequality:
$$x^{2} - 4 x < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x^{2} - 4 x = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = -4$$
$$c = 0$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 0 + \left(-4\right)^{2} = 16$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 4$$
Simplify
$$x_{2} = 0$$
Simplify
$$x_{1} = 4$$
$$x_{2} = 0$$
$$x_{1} = 4$$
$$x_{2} = 0$$
This roots
$$x_{2} = 0$$
$$x_{1} = 4$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$x^{2} - 4 x < 0$$
$$\left(- \frac{1}{10}\right)^{2} - 4 \left(- \frac{1}{10}\right) < 0$$

 41    
--- < 0
100    

but

 41    
--- > 0
100    

Then
$$x < 0$$
no execute
one of the solutions of our inequality is:
$$x > 0 \wedge x < 4$$

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        /     \  
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       x_2      x_1