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ax=4a+14∣x∣+2x−x2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
a*x = 4*a + 14*|x| + 2*x - x2
ax=−x2+(2x+(4a+14∣x∣))a x = - x_{2} + \left(2 x + \left(4 a + 14 \left|{x}\right|\right)\right)
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.
x≥0x \geq 0
o
0≤x∧x<∞0 \leq x \wedge x < \infty
obtenemos la ecuación
ax−4a−14x−2x+x2=0a x - 4 a - 14 x - 2 x + x_{2} = 0
simplificamos, obtenemos
ax−4a−16x+x2=0a x - 4 a - 16 x + x_{2} = 0
la resolución en este intervalo:
x1=4a−x2a−16x_{1} = \frac{4 a - x_{2}}{a - 16}

2.
x<0x < 0
o
−∞<x∧x<0-\infty < x \wedge x < 0
obtenemos la ecuación
ax−4a−2x−14(−x)+x2=0a x - 4 a - 2 x - 14 \left(- x\right) + x_{2} = 0
simplificamos, obtenemos
ax−4a+12x+x2=0a x - 4 a + 12 x + x_{2} = 0
la resolución en este intervalo:
x2=4a−x2a+12x_{2} = \frac{4 a - x_{2}}{a + 12}


Entonces la respuesta definitiva es:
x1=4a−x2a−16x_{1} = \frac{4 a - x_{2}}{a - 16}
x2=4a−x2a+12x_{2} = \frac{4 a - x_{2}}{a + 12}
Gráfica
Respuesta rápida [src] 
         //-x2 + 4*a      -x2 + 4*a     \     //-x2 + 4*a      -x2 + 4*a     \
         ||---------  for --------- >= 0|     ||---------  for --------- >= 0|
x1 = I*im|< -16 + a        -16 + a      | + re|< -16 + a        -16 + a      |
         ||                             |     ||                             |
         \\   nan         otherwise     /     \\   nan         otherwise     /
x1=re⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise)+iim⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise)x_{1} = \operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         //-x2 + 4*a      -x2 + 4*a    \     //-x2 + 4*a      -x2 + 4*a    \
         ||---------  for --------- < 0|     ||---------  for --------- < 0|
x2 = I*im|<  12 + a         12 + a     | + re|<  12 + a         12 + a     |
         ||                            |     ||                            |
         \\   nan         otherwise    /     \\   nan         otherwise    /
x2=re⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise)+iim⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise)x_{2} = \operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
x2 = re(Piecewise(((4*a - x2)/(a + 12, (4*a - x2)/(a + 12) < 0), (nan, True))) + i*im(Piecewise(((4*a - x2)/(a + 12), (4*a - x2)/(a + 12) < 0), (nan, True))))
Suma y producto de raíces [src] 
suma
    //-x2 + 4*a      -x2 + 4*a     \     //-x2 + 4*a      -x2 + 4*a     \       //-x2 + 4*a      -x2 + 4*a    \     //-x2 + 4*a      -x2 + 4*a    \
    ||---------  for --------- >= 0|     ||---------  for --------- >= 0|       ||---------  for --------- < 0|     ||---------  for --------- < 0|
I*im|< -16 + a        -16 + a      | + re|< -16 + a        -16 + a      | + I*im|<  12 + a         12 + a     | + re|<  12 + a         12 + a     |
    ||                             |     ||                             |       ||                            |     ||                            |
    \\   nan         otherwise     /     \\   nan         otherwise     /       \\   nan         otherwise    /     \\   nan         otherwise    /
(re⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise)+iim⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise))+(re⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise)+iim⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise))\left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
    //-x2 + 4*a      -x2 + 4*a     \       //-x2 + 4*a      -x2 + 4*a    \     //-x2 + 4*a      -x2 + 4*a     \     //-x2 + 4*a      -x2 + 4*a    \
    ||---------  for --------- >= 0|       ||---------  for --------- < 0|     ||---------  for --------- >= 0|     ||---------  for --------- < 0|
I*im|< -16 + a        -16 + a      | + I*im|<  12 + a         12 + a     | + re|< -16 + a        -16 + a      | + re|<  12 + a         12 + a     |
    ||                             |       ||                            |     ||                             |     ||                            |
    \\   nan         otherwise     /       \\   nan         otherwise    /     \\   nan         otherwise     /     \\   nan         otherwise    /
re⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise)+re⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise)+iim⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise)+iim⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise)\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
producto
/    //-x2 + 4*a      -x2 + 4*a     \     //-x2 + 4*a      -x2 + 4*a     \\ /    //-x2 + 4*a      -x2 + 4*a    \     //-x2 + 4*a      -x2 + 4*a    \\
|    ||---------  for --------- >= 0|     ||---------  for --------- >= 0|| |    ||---------  for --------- < 0|     ||---------  for --------- < 0||
|I*im|< -16 + a        -16 + a      | + re|< -16 + a        -16 + a      ||*|I*im|<  12 + a         12 + a     | + re|<  12 + a         12 + a     ||
|    ||                             |     ||                             || |    ||                            |     ||                            ||
\    \\   nan         otherwise     /     \\   nan         otherwise     // \    \\   nan         otherwise    /     \\   nan         otherwise    //
(re⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise)+iim⁡({4a−x2a−16for 4a−x2a−16≥0NaNotherwise))(re⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise)+iim⁡({4a−x2a+12for 4a−x2a+12<0NaNotherwise))\left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
/(I*((-16 + re(a))*(-im(x2) + 4*im(a)) - (-re(x2) + 4*re(a))*im(a)) + (-16 + re(a))*(-re(x2) + 4*re(a)) + (-im(x2) + 4*im(a))*im(a))*(I*((12 + re(a))*(-im(x2) + 4*im(a)) - (-re(x2) + 4*re(a))*im(a)) + (12 + re(a))*(-re(x2) + 4*re(a)) + (-im(x2) + 4*im(a))*im(a))         /-x2 + 4*a       -x2 + 4*a    \
|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------  for And|--------- >= 0, --------- < 0|
|                                                                                                          /             2     2   \ /            2     2   \                                                                                                                  \ -16 + a          12 + a     /
<                                                                                                          \(-16 + re(a))  + im (a)/*\(12 + re(a))  + im (a)/                                                                                                                                                 
|                                                                                                                                                                                                                                                                                                             
|                                                                                                                                 nan                                                                                                                                                 otherwise               
\
{(i((re⁡(a)−16)(4im⁡(a)−im⁡(x2))−(4re⁡(a)−re⁡(x2))im⁡(a))+(re⁡(a)−16)(4re⁡(a)−re⁡(x2))+(4im⁡(a)−im⁡(x2))im⁡(a))(i((re⁡(a)+12)(4im⁡(a)−im⁡(x2))−(4re⁡(a)−re⁡(x2))im⁡(a))+(re⁡(a)+12)(4re⁡(a)−re⁡(x2))+(4im⁡(a)−im⁡(x2))im⁡(a))((re⁡(a)−16)2+(im⁡(a))2)((re⁡(a)+12)2+(im⁡(a))2)for 4a−x2a−16≥0∧4a−x2a+12<0NaNotherwise\begin{cases} \frac{\left(i \left(\left(\operatorname{re}{\left(a\right)} - 16\right) \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) - \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right) + \left(\operatorname{re}{\left(a\right)} - 16\right) \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) + \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right) \left(i \left(\left(\operatorname{re}{\left(a\right)} + 12\right) \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) - \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right) + \left(\operatorname{re}{\left(a\right)} + 12\right) \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) + \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right)}{\left(\left(\operatorname{re}{\left(a\right)} - 16\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right) \left(\left(\operatorname{re}{\left(a\right)} + 12\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \wedge \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases} 
Piecewise(((i*((-16 + re(a))*(-im(x2) + 4*im(a)) - (-re(x2) + 4*re(a))*im(a)) + (-16 + re(a))*(-re(x2) + 4*re(a)) + (-im(x2) + 4*im(a))*im(a))*(i*((12 + re(a))*(-im(x2) + 4*im(a)) - (-re(x2) + 4*re(a))*im(a)) + (12 + re(a))*(-re(x2) + 4*re(a)) + (-im(x2) + 4*im(a))*im(a))/(((-16 + re(a))^2 + im(a)^2)*((12 + re(a))^2 + im(a)^2)), ((-x2 + 4*a)/(-16 + a) >= 0)∧((-x2 + 4*a)/(12 + a) < 0)), (nan, True))
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