-4x^2+6*x<=0 inequation

Given the inequality:
$$- 4 x^{2} + 6 x \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 4 x^{2} + 6 x = 0$$
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
$$a = -4$$
$$b = 6$$
$$c = 0$$
, then

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

(6)^2 - 4 * (-4) * (0) = 36

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

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

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

or
$$x_{1} = 0$$
$$x_{2} = \frac{3}{2}$$
$$x_{1} = 0$$
$$x_{2} = \frac{3}{2}$$
$$x_{1} = 0$$
$$x_{2} = \frac{3}{2}$$
This roots
$$x_{1} = 0$$
$$x_{2} = \frac{3}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$- 4 x^{2} + 6 x \leq 0$$
$$\frac{\left(-1\right) 6}{10} - 4 \left(- \frac{1}{10}\right)^{2} \leq 0$$

-16      
---- <= 0
 25      

one of the solutions of our inequality is:
$$x \leq 0$$

 _____           _____          
      \         /
-------•-------•-------
       x1      x2

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 0$$
$$x \geq \frac{3}{2}$$