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.
$$x \geq 0$$
o
$$0 \leq x \wedge x < \infty$$
obtenemos la ecuación
$$a x - 4 a - 14 x - 2 x + x_{2} = 0$$
simplificamos, obtenemos
$$a x - 4 a - 16 x + x_{2} = 0$$
la resolución en este intervalo:
$$x_{1} = \frac{4 a - x_{2}}{a - 16}$$
2.
$$x < 0$$
o
$$-\infty < x \wedge x < 0$$
obtenemos la ecuación
$$a x - 4 a - 2 x - 14 \left(- x\right) + x_{2} = 0$$
simplificamos, obtenemos
$$a x - 4 a + 12 x + x_{2} = 0$$
la resolución en este intervalo:
$$x_{2} = \frac{4 a - x_{2}}{a + 12}$$
Entonces la respuesta definitiva es:
$$x_{1} = \frac{4 a - x_{2}}{a - 16}$$
$$x_{2} = \frac{4 a - x_{2}}{a + 12}$$
//-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 /
$$x_{1} = \operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
//-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 /
$$x_{2} = \operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
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]
//-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 /
$$\left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
//-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 /
$$\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
/ //-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 //
$$\left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a - 16} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{4 a - x_{2}}{a + 12} & \text{for}\: \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
/(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
\
$$\begin{cases} \frac{\left(i \left(\left(\operatorname{re}{\left(a\right)} - 16\right) \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) - \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right) + \left(\operatorname{re}{\left(a\right)} - 16\right) \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) + \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right) \left(i \left(\left(\operatorname{re}{\left(a\right)} + 12\right) \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) - \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right) + \left(\operatorname{re}{\left(a\right)} + 12\right) \left(4 \operatorname{re}{\left(a\right)} - \operatorname{re}{\left(x_{2}\right)}\right) + \left(4 \operatorname{im}{\left(a\right)} - \operatorname{im}{\left(x_{2}\right)}\right) \operatorname{im}{\left(a\right)}\right)}{\left(\left(\operatorname{re}{\left(a\right)} - 16\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right) \left(\left(\operatorname{re}{\left(a\right)} + 12\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} & \text{for}\: \frac{4 a - x_{2}}{a - 16} \geq 0 \wedge \frac{4 a - x_{2}}{a + 12} < 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
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))