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(3x-5)(x-2)=3x+x_4 equation

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Numerical solution:

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The solution

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(3*x - 5)*(x - 2) = 3*x + x_4
$$\left(x - 2\right) \left(3 x - 5\right) = 3 x + x_{4}$$
Simplify
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$\left(x - 2\right) \left(3 x - 5\right) = 3 x + x_{4}$$
to
$$\left(- 3 x - x_{4}\right) + \left(x - 2\right) \left(3 x - 5\right) = 0$$
Expand the expression in the equation
$$\left(- 3 x - x_{4}\right) + \left(x - 2\right) \left(3 x - 5\right) = 0$$
We get the quadratic equation
$$3 x^{2} - 14 x - x_{4} + 10 = 0$$
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
$$a = 3$$
$$b = -14$$
$$c = 10 - x_{4}$$
, then
D = b^2 - 4 * a * c = 

(-14)^2 - 4 * (3) * (10 - x_4) = 76 + 12*x_4

The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = \frac{\sqrt{12 x_{4} + 76}}{6} + \frac{7}{3}$$
$$x_{2} = \frac{7}{3} - \frac{\sqrt{12 x_{4} + 76}}{6}$$
The graph
Sum and product of roots [src] 
sum
       ________________________________                                              ________________________________                                                ________________________________                                              ________________________________                                      
    4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\     4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\       4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\     4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\
    \/  (19 + 3*re(x_4))  + 9*im (x_4) *cos|--------------------------------|   I*\/  (19 + 3*re(x_4))  + 9*im (x_4) *sin|--------------------------------|       \/  (19 + 3*re(x_4))  + 9*im (x_4) *cos|--------------------------------|   I*\/  (19 + 3*re(x_4))  + 9*im (x_4) *sin|--------------------------------|
7                                          \               2                /                                            \               2                /   7                                          \               2                /                                            \               2                /
- - ------------------------------------------------------------------------- - --------------------------------------------------------------------------- + - + ------------------------------------------------------------------------- + ---------------------------------------------------------------------------
3                                       3                                                                            3                                        3                                       3                                                                            3
$$\left(- \frac{i \sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{7}{3}\right) + \left(\frac{i \sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{7}{3}\right)$$ 
=
14/3
$$\frac{14}{3}$$ 
product
/       ________________________________                                              ________________________________                                      \ /       ________________________________                                              ________________________________                                      \
|    4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\     4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\| |    4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\     4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\|
|    \/  (19 + 3*re(x_4))  + 9*im (x_4) *cos|--------------------------------|   I*\/  (19 + 3*re(x_4))  + 9*im (x_4) *sin|--------------------------------|| |    \/  (19 + 3*re(x_4))  + 9*im (x_4) *cos|--------------------------------|   I*\/  (19 + 3*re(x_4))  + 9*im (x_4) *sin|--------------------------------||
|7                                          \               2                /                                            \               2                /| |7                                          \               2                /                                            \               2                /|
|- - ------------------------------------------------------------------------- - ---------------------------------------------------------------------------|*|- + ------------------------------------------------------------------------- + ---------------------------------------------------------------------------|
\3                                       3                                                                            3                                     / \3                                       3                                                                            3                                     /
$$\left(- \frac{i \sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{7}{3}\right) \left(\frac{i \sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{7}{3}\right)$$ 
=
10   re(x_4)   I*im(x_4)
-- - ------- - ---------
3       3          3
$$- \frac{\operatorname{re}{\left(x_{4}\right)}}{3} - \frac{i \operatorname{im}{\left(x_{4}\right)}}{3} + \frac{10}{3}$$ 
10/3 - re(x_4)/3 - i*im(x_4)/3
Rapid solution [src] 
            ________________________________                                              ________________________________                                      
         4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\     4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\
         \/  (19 + 3*re(x_4))  + 9*im (x_4) *cos|--------------------------------|   I*\/  (19 + 3*re(x_4))  + 9*im (x_4) *sin|--------------------------------|
     7                                          \               2                /                                            \               2                /
x1 = - - ------------------------------------------------------------------------- - ---------------------------------------------------------------------------
     3                                       3                                                                            3
$$x_{1} = - \frac{i \sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{7}{3}$$ 
            ________________________________                                              ________________________________                                      
         4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\     4 /                 2       2          /atan2(3*im(x_4), 19 + 3*re(x_4))\
         \/  (19 + 3*re(x_4))  + 9*im (x_4) *cos|--------------------------------|   I*\/  (19 + 3*re(x_4))  + 9*im (x_4) *sin|--------------------------------|
     7                                          \               2                /                                            \               2                /
x2 = - + ------------------------------------------------------------------------- + ---------------------------------------------------------------------------
     3                                       3                                                                            3
$$x_{2} = \frac{i \sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(3 \operatorname{re}{\left(x_{4}\right)} + 19\right)^{2} + 9 \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(x_{4}\right)},3 \operatorname{re}{\left(x_{4}\right)} + 19 \right)}}{2} \right)}}{3} + \frac{7}{3}$$ 
x2 = i*((3*re(x_4) + 19)^2 + 9*im(x_4)^2)^(1/4)*sin(atan2(3*im(x_4, 3*re(x_4) + 19)/2)/3 + ((3*re(x_4) + 19)^2 + 9*im(x_4)^2)^(1/4)*cos(atan2(3*im(x_4), 3*re(x_4) + 19)/2)/3 + 7/3)
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