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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
∣(x2−6x)+8∣=a\left|{\left(x^{2} - 6 x\right) + 8}\right| = a
Simplificar
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.
x2−6x+8≥0x^{2} - 6 x + 8 \geq 0
o
(4≤x∧x<∞)∨(x≤2∧−∞<x)\left(4 \leq x \wedge x < \infty\right) \vee \left(x \leq 2 \wedge -\infty < x\right)
obtenemos la ecuación
−a+(x2−6x+8)=0- a + \left(x^{2} - 6 x + 8\right) = 0
simplificamos, obtenemos
−a+x2−6x+8=0- a + x^{2} - 6 x + 8 = 0
la resolución en este intervalo:
x1=3−a+1x_{1} = 3 - \sqrt{a + 1}
x2=a+1+3x_{2} = \sqrt{a + 1} + 3

2.
x2−6x+8<0x^{2} - 6 x + 8 < 0
o
2<x∧x<42 < x \wedge x < 4
obtenemos la ecuación
−a+(−x2+6x−8)=0- a + \left(- x^{2} + 6 x - 8\right) = 0
simplificamos, obtenemos
−a−x2+6x−8=0- a - x^{2} + 6 x - 8 = 0
la resolución en este intervalo:
x3=3−1−ax_{3} = 3 - \sqrt{1 - a}
x4=1−a+3x_{4} = \sqrt{1 - a} + 3


Entonces la respuesta definitiva es:
x1=3−a+1x_{1} = 3 - \sqrt{a + 1}
x2=a+1+3x_{2} = \sqrt{a + 1} + 3
x3=3−1−ax_{3} = 3 - \sqrt{1 - a}
x4=1−a+3x_{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                  /
x1=re⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise)+iim⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise)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                   /
x2=re⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)+iim⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)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                 /
x3=re⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)+iim⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)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                  /
x4=re⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise)+iim⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise)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                  /
(((re⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise)+iim⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise))+(re⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)+iim⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)))+(re⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)+iim⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)))+(re⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise)+iim⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise))\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                  /
re⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise)+re⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)+re⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)+re⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise)+iim⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise)+iim⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)+iim⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)+iim⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise)\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                  //
(re⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise)+iim⁡({3−1−afor 61−a+(1−a−3)2−10<0NaNotherwise))(re⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise)+iim⁡({3−a+1for 6a+1+(a+1−3)2−10≥0NaNotherwise))(re⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise)+iim⁡({1−a+3for 61−a−(1−a+3)2+10>0NaNotherwise))(re⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise)+iim⁡({a+1+3for 6a+1−(a+1+3)2+10≤0NaNotherwise))\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
\
{−(re⁡(a))2−2ire⁡(a)im⁡(a)+(im⁡(a))2+64for a>0NaNotherwise\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))
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