x^2>25 inequation

Given the inequality:
$$x^{2} > 25$$
To solve this inequality, we must first solve the corresponding equation:
$$x^{2} = 25$$
Solve:
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$x^{2} = 25$$
to
$$x^{2} - 25 = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = -25$$
, then
$$D = b^2 - 4 * a * c = $$
$$0^{2} - 1 \cdot 4 \left(-25\right) = 100$$
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
$$x_{1} = 5$$
Simplify
$$x_{2} = -5$$
Simplify
$$x_{1} = 5$$
$$x_{2} = -5$$
$$x_{1} = 5$$
$$x_{2} = -5$$
This roots
$$x_{2} = -5$$
$$x_{1} = 5$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-5 - \frac{1}{10}$$
=
$$- \frac{51}{10}$$
substitute to the expression
$$x^{2} > 25$$
$$\left(- \frac{51}{10}\right)^{2} > 25$$

2601     
---- > 25
100      

one of the solutions of our inequality is:
$$x < -5$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -5$$
$$x > 5$$