√(8x+1)-√(3+x)=√(3x-2) equation

Given the equation
$$- \sqrt{x + 3} + \sqrt{8 x + 1} = \sqrt{3 x - 2}$$
We raise the equation sides to 2-th degree
$$\left(- \sqrt{x + 3} + \sqrt{8 x + 1}\right)^{2} = 3 x - 2$$
or
$$1^{2} \cdot \left(8 x + 1\right) + \left(\left(-1\right) 2 \cdot 1 \sqrt{\left(3 x - 2\right) \left(8 x + 1\right)} + \left(-1\right)^{2} \cdot \left(3 x - 2\right)\right) = 3 x - 2$$
or
$$11 x - 2 \sqrt{24 x^{2} - 13 x - 2} - 1 = 3 x - 2$$
transform:
$$- 2 \sqrt{24 x^{2} - 13 x - 2} = - 8 x - 1$$
We raise the equation sides to 2-th degree
$$96 x^{2} - 52 x - 8 = \left(- 8 x - 1\right)^{2}$$
$$96 x^{2} - 52 x - 8 = 64 x^{2} + 16 x + 1$$
Transfer the right side of the equation left part with negative sign
$$32 x^{2} - 68 x - 9 = 0$$
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
$$a = 32$$
$$b = -68$$
$$c = -9$$
, then

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

(-68)^2 - 4 * (32) * (-9) = 5776

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

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

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

or
$$x_{1} = \frac{9}{4}$$
Simplify
$$x_{2} = - \frac{1}{8}$$
Simplify

Because
$$\sqrt{24 x^{2} - 13 x - 2} = 4 x + \frac{1}{2}$$
and
$$\sqrt{24 x^{2} - 13 x - 2} \geq 0$$
then
$$4 x + \frac{1}{2} \geq 0$$
or
$$- \frac{1}{8} \leq x$$
$$x < \infty$$
$$x_{1} = \frac{9}{4}$$
$$x_{2} = - \frac{1}{8}$$
check:
$$x_{1} = \frac{9}{4}$$
$$- \sqrt{x_{1} + 3} - \sqrt{3 x_{1} - 2} + \sqrt{8 x_{1} + 1} = 0$$
=
$$- \sqrt{\left(-1\right) 2 + 3 \cdot \frac{9}{4}} + \left(- \sqrt{\frac{9}{4} + 3} + \sqrt{1 + 8 \cdot \frac{9}{4}}\right) = 0$$
=

sqrt(19) - sqrt(27/4 - 1*2) - sqrt(21)/2 = 0

- No
$$x_{2} = - \frac{1}{8}$$
$$- \sqrt{x_{2} + 3} - \sqrt{3 x_{2} - 2} + \sqrt{8 x_{2} + 1} = 0$$
=
$$\left(- \sqrt{- \frac{1}{8} + 3} + \sqrt{8 \left(- \frac{1}{8}\right) + 1}\right) - \sqrt{\left(-1\right) 2 + 3 \left(- \frac{1}{8}\right)} = 0$$
=

-sqrt(-3/8 - 1*2) - sqrt(46)/4 = 0

- No
The final answer:
This equation has no roots