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:
$$x_{1} = 3 - \sqrt{a + 1}$$
$$x_{2} = \sqrt{a + 1} + 3$$
$$x_{3} = 3 - \sqrt{1 - a}$$
$$x_{4} = \sqrt{1 - a} + 3$$
// 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]
// 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 /
$$\operatorname{re}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{1 - a} & \text{for}\: 6 \sqrt{1 - a} + \left(\sqrt{1 - a} - 3\right)^{2} - 10 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 3 - \sqrt{a + 1} & \text{for}\: 6 \sqrt{a + 1} + \left(\sqrt{a + 1} - 3\right)^{2} - 10 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{1 - a} + 3 & \text{for}\: 6 \sqrt{1 - a} - \left(\sqrt{1 - a} + 3\right)^{2} + 10 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 1} + 3 & \text{for}\: 6 \sqrt{a + 1} - \left(\sqrt{a + 1} + 3\right)^{2} + 10 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
/ // 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))