36x^2+12x+1<0 inequation

Given the inequality:
$$\left(36 x^{2} + 12 x\right) + 1 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(36 x^{2} + 12 x\right) + 1 = 0$$
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
$$a = 36$$
$$b = 12$$
$$c = 1$$
, then

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

(12)^2 - 4 * (36) * (1) = 0

Because D = 0, then the equation has one root.

x = -b/2a = -12/2/(36)

$$x_{1} = - \frac{1}{6}$$
$$x_{1} = - \frac{1}{6}$$
$$x_{1} = - \frac{1}{6}$$
This roots
$$x_{1} = - \frac{1}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{6} + - \frac{1}{10}$$
=
$$- \frac{4}{15}$$
substitute to the expression
$$\left(36 x^{2} + 12 x\right) + 1 < 0$$
$$\left(\frac{\left(-4\right) 12}{15} + 36 \left(- \frac{4}{15}\right)^{2}\right) + 1 < 0$$

9/25 < 0

but

9/25 > 0

Then
$$x < - \frac{1}{6}$$
no execute
the solution of our inequality is:
$$x > - \frac{1}{6}$$

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