5x^2-4x-1<0 inequation

Given the inequality:
$$5 x^{2} - 4 x - 1 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$5 x^{2} - 4 x - 1 = 0$$
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
$$a = 5$$
$$b = -4$$
$$c = -1$$
, then

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

(-4)^2 - 4 * (5) * (-1) = 36

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

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

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

or
$$x_{1} = 1$$
Simplify
$$x_{2} = - \frac{1}{5}$$
Simplify
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{5}$$
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{5}$$
This roots
$$x_{2} = - \frac{1}{5}$$
$$x_{1} = 1$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{5} - \frac{1}{10}$$
=
$$- \frac{3}{10}$$
substitute to the expression
$$5 x^{2} - 4 x - 1 < 0$$
$$\left(-1\right) 1 + 5 \left(- \frac{3}{10}\right)^{2} - 4 \left(- \frac{3}{10}\right) < 0$$

13    
-- < 0
20    

but

13    
-- > 0
20    

Then
$$x < - \frac{1}{5}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{1}{5} \wedge x < 1$$

         _____  
        /     \  
-------ο-------ο-------
       x_2      x_1