(x-3)√x^2-√1<0 inequation

Given the inequality:
$$\left(x - 3\right) \left(\sqrt{x}\right)^{2} - \sqrt{1} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 3\right) \left(\sqrt{x}\right)^{2} - \sqrt{1} = 0$$
Solve:
Expand the expression in the equation
$$\left(x - 3\right) \left(\sqrt{x}\right)^{2} - \sqrt{1} = 0$$
We get the quadratic equation
$$- 3 x + x x - 1 = 0$$
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
$$a = 1$$
$$b = -3$$
$$c = -1$$
, then

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

(-3)^2 - 4 * (1) * (-1) = 13

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

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

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

or
$$x_{1} = \frac{3}{2} + \frac{\sqrt{13}}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{13}}{2}$$
$$x_{1} = \frac{3}{2} + \frac{\sqrt{13}}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{13}}{2}$$
$$x_{1} = \frac{3}{2} + \frac{\sqrt{13}}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{13}}{2}$$
This roots
$$x_{2} = \frac{3}{2} - \frac{\sqrt{13}}{2}$$
$$x_{1} = \frac{3}{2} + \frac{\sqrt{13}}{2}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(\frac{3}{2} - \frac{\sqrt{13}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{7}{5} - \frac{\sqrt{13}}{2}$$
substitute to the expression
$$\left(x - 3\right) \left(\sqrt{x}\right)^{2} - \sqrt{1} < 0$$
$$- \sqrt{1} + \left(-3 + \left(\frac{7}{5} - \frac{\sqrt{13}}{2}\right)\right) \left(\sqrt{\frac{7}{5} - \frac{\sqrt{13}}{2}}\right)^{2} < 0$$

     /        ____\ /      ____\    
     |  8   \/ 13 | |7   \/ 13 |    
-1 + |- - - ------|*|- - ------| < 0
     \  5     2   / \5     2   /    
    

but

     /        ____\ /      ____\    
     |  8   \/ 13 | |7   \/ 13 |    
-1 + |- - - ------|*|- - ------| > 0
     \  5     2   / \5     2   /    
    

Then
$$x < \frac{3}{2} - \frac{\sqrt{13}}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{3}{2} - \frac{\sqrt{13}}{2} \wedge x < \frac{3}{2} + \frac{\sqrt{13}}{2}$$

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