22-x²>1/4 inequation

Given the inequality:
$$22 - x^{2} > \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$22 - x^{2} = \frac{1}{4}$$
Solve:
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$22 - x^{2} = \frac{1}{4}$$
to
$$\left(22 - x^{2}\right) - \frac{1}{4} = 0$$
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
$$a = -1$$
$$b = 0$$
$$c = \frac{87}{4}$$
, then

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

(0)^2 - 4 * (-1) * (87/4) = 87

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

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

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

or
$$x_{1} = - \frac{\sqrt{87}}{2}$$
$$x_{2} = \frac{\sqrt{87}}{2}$$
$$x_{1} = - \frac{\sqrt{87}}{2}$$
$$x_{2} = \frac{\sqrt{87}}{2}$$
$$x_{1} = - \frac{\sqrt{87}}{2}$$
$$x_{2} = \frac{\sqrt{87}}{2}$$
This roots
$$x_{1} = - \frac{\sqrt{87}}{2}$$
$$x_{2} = \frac{\sqrt{87}}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\sqrt{87}}{2} - \frac{1}{10}$$
=
$$- \frac{\sqrt{87}}{2} - \frac{1}{10}$$
substitute to the expression
$$22 - x^{2} > \frac{1}{4}$$
$$22 - \left(- \frac{\sqrt{87}}{2} - \frac{1}{10}\right)^{2} > \frac{1}{4}$$

                    2      
     /         ____\       
     |  1    \/ 87 |  > 1/4
22 - |- -- - ------|       
     \  10     2   /       

Then
$$x < - \frac{\sqrt{87}}{2}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\sqrt{87}}{2} \wedge x < \frac{\sqrt{87}}{2}$$

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