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.
$$x^{2} - 2 \geq 0$$
o
$$\left(x \leq - \sqrt{2} \wedge -\infty < x\right) \vee \left(\sqrt{2} \leq x \wedge x < \infty\right)$$
obtenemos la ecuación
$$- a + \left(x^{2} - 2\right) = 0$$
simplificamos, obtenemos
$$- a + x^{2} - 2 = 0$$
la resolución en este intervalo:
$$x_{1} = - \sqrt{a + 2}$$
$$x_{2} = \sqrt{a + 2}$$
2.
$$x^{2} - 2 < 0$$
o
$$- \sqrt{2} < x \wedge x < \sqrt{2}$$
obtenemos la ecuación
$$- a + \left(2 - x^{2}\right) = 0$$
simplificamos, obtenemos
$$- a - x^{2} + 2 = 0$$
la resolución en este intervalo:
$$x_{3} = - \sqrt{2 - a}$$
$$x_{4} = \sqrt{2 - a}$$
Entonces la respuesta definitiva es:
$$x_{1} = - \sqrt{a + 2}$$
$$x_{2} = \sqrt{a + 2}$$
$$x_{3} = - \sqrt{2 - a}$$
$$x_{4} = \sqrt{2 - a}$$
Suma y producto de raíces
[src]
// _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \
||-\/ 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 |
\\ / \\ / \\ / \\ / \\ / \\ / \\ / \\ /
$$\left(\left(\left(\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
// _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \
||\/ 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|
\\ / \\ / \\ / \\ / \\ / \\ / \\ / \\ /
$$\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
/ // _______ \ // _______ \\ / // _______ \ // _______ \\ / // _______ \ // _______ \\ / // _______ \ // _______ \\
| ||-\/ 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 ||
\ \\ / \\ // \ \\ / \\ // \ \\ / \\ // \ \\ / \\ //
$$\left(\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
/ ________________________ _______________________
| / 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
$$\begin{cases} \sqrt{\left(\operatorname{re}{\left(a\right)} - 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sqrt{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} e^{i \left(\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},2 - \operatorname{re}{\left(a\right)} \right)} + \operatorname{atan_{2}}{\left(\operatorname{im}{\left(a\right)},\operatorname{re}{\left(a\right)} + 2 \right)}\right)} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
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))
// _______ \ // _______ \
||-\/ 2 - a for a > 0| ||-\/ 2 - a for a > 0|
x1 = I*im|< | + re|< |
|| nan otherwise| || nan otherwise|
\\ / \\ /
$$x_{1} = \operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// _______ \ // _______ \
||\/ 2 - a for a > 0| ||\/ 2 - a for a > 0|
x2 = I*im|< | + re|< |
|| nan otherwise| || nan otherwise|
\\ / \\ /
$$x_{2} = \operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// _______ \ // _______ \
||-\/ 2 + a for a >= 0| ||-\/ 2 + a for a >= 0|
x3 = I*im|< | + re|< |
|| nan otherwise | || nan otherwise |
\\ / \\ /
$$x_{3} = \operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// _______ \ // _______ \
||\/ 2 + a for a >= 0| ||\/ 2 + a for a >= 0|
x4 = I*im|< | + re|< |
|| nan otherwise | || nan otherwise |
\\ / \\ /
$$x_{4} = \operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
x4 = re(Piecewise((sqrt(a + 2, a >= 0), (nan, True))) + i*im(Piecewise((sqrt(a + 2), a >= 0), (nan, True))))