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|x^2-6x+8|=а la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
| 2          |    
|x  - 6*x + 8| = a
$$\left|{\left(x^{2} - 6 x\right) + 8}\right| = a$$
Solución detallada
Para cada expresión dentro del módulo en la ecuación
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.

1.
$$x^{2} - 6 x + 8 \geq 0$$
o
$$\left(4 \leq x \wedge x < \infty\right) \vee \left(x \leq 2 \wedge -\infty < x\right)$$
obtenemos la ecuación
$$- a + \left(x^{2} - 6 x + 8\right) = 0$$
simplificamos, obtenemos
$$- a + x^{2} - 6 x + 8 = 0$$
la resolución en este intervalo:
$$x_{1} = 3 - \sqrt{a + 1}$$
$$x_{2} = \sqrt{a + 1} + 3$$

2.
$$x^{2} - 6 x + 8 < 0$$
o
$$2 < x \wedge x < 4$$
obtenemos la ecuación
$$- a + \left(- x^{2} + 6 x - 8\right) = 0$$
simplificamos, obtenemos
$$- a - x^{2} + 6 x - 8 = 0$$
la resolución en este intervalo:
$$x_{3} = 3 - \sqrt{1 - a}$$
$$x_{4} = \sqrt{1 - a} + 3$$


Entonces la respuesta definitiva es:
$$x_{1} = 3 - \sqrt{a + 1}$$
$$x_{2} = \sqrt{a + 1} + 3$$
$$x_{3} = 3 - \sqrt{1 - a}$$
$$x_{4} = \sqrt{1 - a} + 3$$
Gráfica
Respuesta rápida [src]
         //                                         2                  \     //                                         2                  \
         ||      _______            /       _______\        _______    |     ||      _______            /       _______\        _______    |
x1 = I*im|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0| + re|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0|
         ||                                                            |     ||                                                            |
         \\     nan                         otherwise                  /     \\     nan                         otherwise                  /
$$x_{1} = \operatorname{re}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
         //                                         2                   \     //                                         2                   \
         ||      _______            /       _______\        _______     |     ||      _______            /       _______\        _______     |
x2 = I*im|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0| + re|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0|
         ||                                                             |     ||                                                             |
         \\     nan                         otherwise                   /     \\     nan                         otherwise                   /
$$x_{2} = \operatorname{re}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
         //                                       2                  \     //                                       2                  \
         ||      _______           /      _______\        _______    |     ||      _______           /      _______\        _______    |
x3 = I*im|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0| + re|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0|
         ||                                                          |     ||                                                          |
         \\     nan                        otherwise                 /     \\     nan                        otherwise                 /
$$x_{3} = \operatorname{re}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
         //                                       2                   \     //                                       2                   \
         ||      _______           /      _______\        _______     |     ||      _______           /      _______\        _______     |
x4 = I*im|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0| + re|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0|
         ||                                                           |     ||                                                           |
         \\     nan                        otherwise                  /     \\     nan                        otherwise                  /
$$x_{4} = \operatorname{re}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
x4 = re(Piecewise((sqrt(a + 1 + 3, 6*sqrt(a + 1) - (sqrt(a + 1) + 3)^2 + 10 <= 0), (nan, True))) + i*im(Piecewise((sqrt(a + 1) + 3, 6*sqrt(a + 1) - (sqrt(a + 1) + 3)^2 + 10 <= 0), (nan, True))))
Suma y producto de raíces [src]
suma
    //                                         2                  \     //                                         2                  \       //                                         2                   \     //                                         2                   \       //                                       2                  \     //                                       2                  \       //                                       2                   \     //                                       2                   \
    ||      _______            /       _______\        _______    |     ||      _______            /       _______\        _______    |       ||      _______            /       _______\        _______     |     ||      _______            /       _______\        _______     |       ||      _______           /      _______\        _______    |     ||      _______           /      _______\        _______    |       ||      _______           /      _______\        _______     |     ||      _______           /      _______\        _______     |
I*im|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0| + re|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0| + I*im|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0| + re|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0| + I*im|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0| + re|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0| + I*im|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0| + re|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0|
    ||                                                            |     ||                                                            |       ||                                                             |     ||                                                             |       ||                                                          |     ||                                                          |       ||                                                           |     ||                                                           |
    \\     nan                         otherwise                  /     \\     nan                         otherwise                  /       \\     nan                         otherwise                   /     \\     nan                         otherwise                   /       \\     nan                        otherwise                 /     \\     nan                        otherwise                 /       \\     nan                        otherwise                  /     \\     nan                        otherwise                  /
$$\left(\left(\left(\operatorname{re}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
    //                                       2                   \       //                                       2                  \       //                                         2                   \       //                                         2                  \     //                                       2                   \     //                                       2                  \     //                                         2                   \     //                                         2                  \
    ||      _______           /      _______\        _______     |       ||      _______           /      _______\        _______    |       ||      _______            /       _______\        _______     |       ||      _______            /       _______\        _______    |     ||      _______           /      _______\        _______     |     ||      _______           /      _______\        _______    |     ||      _______            /       _______\        _______     |     ||      _______            /       _______\        _______    |
I*im|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0| + I*im|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0| + I*im|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0| + I*im|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0| + re|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0| + re|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0| + re|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0| + re|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0|
    ||                                                           |       ||                                                          |       ||                                                             |       ||                                                            |     ||                                                           |     ||                                                          |     ||                                                             |     ||                                                            |
    \\     nan                        otherwise                  /       \\     nan                        otherwise                 /       \\     nan                         otherwise                   /       \\     nan                         otherwise                  /     \\     nan                        otherwise                  /     \\     nan                        otherwise                 /     \\     nan                         otherwise                   /     \\     nan                         otherwise                  /
$$\operatorname{re}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
producto
/    //                                         2                  \     //                                         2                  \\ /    //                                         2                   \     //                                         2                   \\ /    //                                       2                  \     //                                       2                  \\ /    //                                       2                   \     //                                       2                   \\
|    ||      _______            /       _______\        _______    |     ||      _______            /       _______\        _______    || |    ||      _______            /       _______\        _______     |     ||      _______            /       _______\        _______     || |    ||      _______           /      _______\        _______    |     ||      _______           /      _______\        _______    || |    ||      _______           /      _______\        _______     |     ||      _______           /      _______\        _______     ||
|I*im|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0| + re|<3 - \/ 1 - a   for -10 + \-3 + \/ 1 - a /  + 6*\/ 1 - a  < 0||*|I*im|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0| + re|<3 - \/ 1 + a   for -10 + \-3 + \/ 1 + a /  + 6*\/ 1 + a  >= 0||*|I*im|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0| + re|<3 + \/ 1 - a   for 10 - \3 + \/ 1 - a /  + 6*\/ 1 - a  > 0||*|I*im|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0| + re|<3 + \/ 1 + a   for 10 - \3 + \/ 1 + a /  + 6*\/ 1 + a  <= 0||
|    ||                                                            |     ||                                                            || |    ||                                                             |     ||                                                             || |    ||                                                          |     ||                                                          || |    ||                                                           |     ||                                                           ||
\    \\     nan                         otherwise                  /     \\     nan                         otherwise                  // \    \\     nan                         otherwise                   /     \\     nan                         otherwise                   // \    \\     nan                        otherwise                 /     \\     nan                        otherwise                 // \    \\     nan                        otherwise                  /     \\     nan                        otherwise                  //
$$\left(\operatorname{re}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
/       2        2                                
|64 + im (a) - re (a) - 2*I*im(a)*re(a)  for a > 0
<                                                 
|                 nan                    otherwise
\                                                 
$$\begin{cases} - \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 64 & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
Piecewise((64 + im(a)^2 - re(a)^2 - 2*i*im(a)*re(a), a > 0), (nan, True))