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absolute(x^2-2)=a la ecuación

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Solución numérica:

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Solución

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

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


Entonces la respuesta definitiva es:
x1=−a+2x_{1} = - \sqrt{a + 2}
x2=a+2x_{2} = \sqrt{a + 2}
x3=−2−ax_{3} = - \sqrt{2 - a}
x4=2−ax_{4} = \sqrt{2 - a}
Gráfica
Suma y producto de raíces [src] 
suma
    //   _______           \     //   _______           \       //  _______           \     //  _______           \       //   _______            \     //   _______            \       //  _______            \     //  _______            \
    ||-\/ 2 - a   for a > 0|     ||-\/ 2 - a   for a > 0|       ||\/ 2 - a   for a > 0|     ||\/ 2 - a   for a > 0|       ||-\/ 2 + a   for a >= 0|     ||-\/ 2 + a   for a >= 0|       ||\/ 2 + a   for a >= 0|     ||\/ 2 + a   for a >= 0|
I*im|<                     | + re|<                     | + I*im|<                    | + re|<                    | + I*im|<                      | + re|<                      | + I*im|<                     | + re|<                     |
    ||   nan      otherwise|     ||   nan      otherwise|       ||   nan     otherwise|     ||   nan     otherwise|       ||   nan      otherwise |     ||   nan      otherwise |       ||   nan     otherwise |     ||   nan     otherwise |
    \\                     /     \\                     /       \\                    /     \\                    /       \\                      /     \\                      /       \\                     /     \\                     /
(((re⁡({−2−afor a>0NaNotherwise)+iim⁡({−2−afor a>0NaNotherwise))+(re⁡({2−afor a>0NaNotherwise)+iim⁡({2−afor a>0NaNotherwise)))+(re⁡({−a+2for a≥0NaNotherwise)+iim⁡({−a+2for a≥0NaNotherwise)))+(re⁡({a+2for a≥0NaNotherwise)+iim⁡({a+2for a≥0NaNotherwise))\left(\left(\left(\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
    //  _______            \       //  _______           \       //   _______            \       //   _______           \     //  _______            \     //  _______           \     //   _______            \     //   _______           \
    ||\/ 2 + a   for a >= 0|       ||\/ 2 - a   for a > 0|       ||-\/ 2 + a   for a >= 0|       ||-\/ 2 - a   for a > 0|     ||\/ 2 + a   for a >= 0|     ||\/ 2 - a   for a > 0|     ||-\/ 2 + a   for a >= 0|     ||-\/ 2 - a   for a > 0|
I*im|<                     | + I*im|<                    | + I*im|<                      | + I*im|<                     | + re|<                     | + re|<                    | + re|<                      | + re|<                     |
    ||   nan     otherwise |       ||   nan     otherwise|       ||   nan      otherwise |       ||   nan      otherwise|     ||   nan     otherwise |     ||   nan     otherwise|     ||   nan      otherwise |     ||   nan      otherwise|
    \\                     /       \\                    /       \\                      /       \\                     /     \\                     /     \\                    /     \\                      /     \\                     /
re⁡({−2−afor a>0NaNotherwise)+re⁡({2−afor a>0NaNotherwise)+re⁡({−a+2for a≥0NaNotherwise)+re⁡({a+2for a≥0NaNotherwise)+iim⁡({−2−afor a>0NaNotherwise)+iim⁡({2−afor a>0NaNotherwise)+iim⁡({−a+2for a≥0NaNotherwise)+iim⁡({a+2for a≥0NaNotherwise)\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
producto
/    //   _______           \     //   _______           \\ /    //  _______           \     //  _______           \\ /    //   _______            \     //   _______            \\ /    //  _______            \     //  _______            \\
|    ||-\/ 2 - a   for a > 0|     ||-\/ 2 - a   for a > 0|| |    ||\/ 2 - a   for a > 0|     ||\/ 2 - a   for a > 0|| |    ||-\/ 2 + a   for a >= 0|     ||-\/ 2 + a   for a >= 0|| |    ||\/ 2 + a   for a >= 0|     ||\/ 2 + a   for a >= 0||
|I*im|<                     | + re|<                     ||*|I*im|<                    | + re|<                    ||*|I*im|<                      | + re|<                      ||*|I*im|<                     | + re|<                     ||
|    ||   nan      otherwise|     ||   nan      otherwise|| |    ||   nan     otherwise|     ||   nan     otherwise|| |    ||   nan      otherwise |     ||   nan      otherwise || |    ||   nan     otherwise |     ||   nan     otherwise ||
\    \\                     /     \\                     // \    \\                    /     \\                    // \    \\                      /     \\                      // \    \\                     /     \\                     //
(re⁡({−2−afor a>0NaNotherwise)+iim⁡({−2−afor a>0NaNotherwise))(re⁡({2−afor a>0NaNotherwise)+iim⁡({2−afor a>0NaNotherwise))(re⁡({−a+2for a≥0NaNotherwise)+iim⁡({−a+2for a≥0NaNotherwise))(re⁡({a+2for a≥0NaNotherwise)+iim⁡({a+2for a≥0NaNotherwise))\left(\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
/   ________________________    _______________________                                                                   
|  /             2     2       /            2     2      I*(atan2(-im(a), 2 - re(a)) + atan2(im(a), 2 + re(a)))           
<\/  (-2 + re(a))  + im (a) *\/  (2 + re(a))  + im (a) *e                                                        for a > 0
|                                                                                                                         
\                                                     nan                                                        otherwise
{(re⁡(a)−2)2+(im⁡(a))2(re⁡(a)+2)2+(im⁡(a))2ei(atan2⁡(−im⁡(a),2−re⁡(a))+atan2⁡(im⁡(a),re⁡(a)+2))for a>0NaNotherwise\begin{cases} \sqrt{\left(\operatorname{re}{\left(a\right)} - 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sqrt{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} e^{i \left(\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},2 - \operatorname{re}{\left(a\right)} \right)} + \operatorname{atan_{2}}{\left(\operatorname{im}{\left(a\right)},\operatorname{re}{\left(a\right)} + 2 \right)}\right)} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases} 
Piecewise((sqrt((-2 + re(a))^2 + im(a)^2)*sqrt((2 + re(a))^2 + im(a)^2)*exp(i*(atan2(-im(a), 2 - re(a)) + atan2(im(a), 2 + re(a)))), a > 0), (nan, True))
Respuesta rápida [src] 
         //   _______           \     //   _______           \
         ||-\/ 2 - a   for a > 0|     ||-\/ 2 - a   for a > 0|
x1 = I*im|<                     | + re|<                     |
         ||   nan      otherwise|     ||   nan      otherwise|
         \\                     /     \\                     /
x1=re⁡({−2−afor a>0NaNotherwise)+iim⁡({−2−afor a>0NaNotherwise)x_{1} = \operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //  _______           \     //  _______           \
         ||\/ 2 - a   for a > 0|     ||\/ 2 - a   for a > 0|
x2 = I*im|<                    | + re|<                    |
         ||   nan     otherwise|     ||   nan     otherwise|
         \\                    /     \\                    /
x2=re⁡({2−afor a>0NaNotherwise)+iim⁡({2−afor a>0NaNotherwise)x_{2} = \operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //   _______            \     //   _______            \
         ||-\/ 2 + a   for a >= 0|     ||-\/ 2 + a   for a >= 0|
x3 = I*im|<                      | + re|<                      |
         ||   nan      otherwise |     ||   nan      otherwise |
         \\                      /     \\                      /
x3=re⁡({−a+2for a≥0NaNotherwise)+iim⁡({−a+2for a≥0NaNotherwise)x_{3} = \operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //  _______            \     //  _______            \
         ||\/ 2 + a   for a >= 0|     ||\/ 2 + a   for a >= 0|
x4 = I*im|<                     | + re|<                     |
         ||   nan     otherwise |     ||   nan     otherwise |
         \\                     /     \\                     /
x4=re⁡({a+2for a≥0NaNotherwise)+iim⁡({a+2for a≥0NaNotherwise)x_{4} = \operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
x4 = re(Piecewise((sqrt(a + 2, a >= 0), (nan, True))) + i*im(Piecewise((sqrt(a + 2), a >= 0), (nan, True))))
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