Solución detallada
Tenemos una ecuación lineal:
(a^2-5*a+6)*x = a^2+2*a-8
Abrimos los paréntesis en el miembro izquierdo de la ecuación
a+2-5*a+6x = a^2+2*a-8
Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
x*(6 + a^2 - 5*a) = a^2+2*a-8
Dividamos ambos miembros de la ecuación en 6 + a^2 - 5*a
x = -8 + a^2 + 2*a / (6 + a^2 - 5*a)
Obtenemos la respuesta: x = (4 + a)/(-3 + a)
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$x \left(a^{2} - 5 a + 6\right) = a^{2} + 2 a - 8$$
Коэффициент при x равен
$$a^{2} - 5 a + 6$$
entonces son posibles los casos para a :
$$a < 2$$
$$a = 2$$
$$a > 2 \wedge a < 3$$
$$a = 3$$
Consideremos todos los casos con detalles:
Con
$$a < 2$$
la ecuación será
$$2 x + 5 = 0$$
su solución
$$x = - \frac{5}{2}$$
Con
$$a = 2$$
la ecuación será
$$0 = 0$$
su solución
cualquiera x
Con
$$a > 2 \wedge a < 3$$
la ecuación será
$$- \frac{x}{4} - \frac{13}{4} = 0$$
su solución
$$x = -13$$
Con
$$a = 3$$
la ecuación será
$$-7 = 0$$
su solución
no hay soluciones
2
/ (-3 + re(a))*im(a) (4 + re(a))*im(a) \ im (a) (-3 + re(a))*(4 + re(a))
x1 = I*|---------------------- - ----------------------| + ---------------------- + ------------------------
| 2 2 2 2 | 2 2 2 2
\(-3 + re(a)) + im (a) (-3 + re(a)) + im (a)/ (-3 + re(a)) + im (a) (-3 + re(a)) + im (a)
$$x_{1} = i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \left(\operatorname{re}{\left(a\right)} + 4\right)}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
x1 = i*((re(a) - 3)*im(a)/((re(a) - 3)^2 + im(a)^2) - (re(a) + 4)*im(a)/((re(a) - 3)^2 + im(a)^2)) + (re(a) - 3)*(re(a) + 4)/((re(a) - 3)^2 + im(a)^2) + im(a)^2/((re(a) - 3)^2 + im(a)^2)
Suma y producto de raíces
[src]
2
/ (-3 + re(a))*im(a) (4 + re(a))*im(a) \ im (a) (-3 + re(a))*(4 + re(a))
I*|---------------------- - ----------------------| + ---------------------- + ------------------------
| 2 2 2 2 | 2 2 2 2
\(-3 + re(a)) + im (a) (-3 + re(a)) + im (a)/ (-3 + re(a)) + im (a) (-3 + re(a)) + im (a)
$$i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \left(\operatorname{re}{\left(a\right)} + 4\right)}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
2
/ (-3 + re(a))*im(a) (4 + re(a))*im(a) \ im (a) (-3 + re(a))*(4 + re(a))
I*|---------------------- - ----------------------| + ---------------------- + ------------------------
| 2 2 2 2 | 2 2 2 2
\(-3 + re(a)) + im (a) (-3 + re(a)) + im (a)/ (-3 + re(a)) + im (a) (-3 + re(a)) + im (a)
$$i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \left(\operatorname{re}{\left(a\right)} + 4\right)}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
2
/ (-3 + re(a))*im(a) (4 + re(a))*im(a) \ im (a) (-3 + re(a))*(4 + re(a))
I*|---------------------- - ----------------------| + ---------------------- + ------------------------
| 2 2 2 2 | 2 2 2 2
\(-3 + re(a)) + im (a) (-3 + re(a)) + im (a)/ (-3 + re(a)) + im (a) (-3 + re(a)) + im (a)
$$i \left(\frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\operatorname{re}{\left(a\right)} + 4\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \left(\operatorname{re}{\left(a\right)} + 4\right)}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
2
im (a) + (-3 + re(a))*(4 + re(a)) - 7*I*im(a)
---------------------------------------------
2 2
(-3 + re(a)) + im (a)
$$\frac{\left(\operatorname{re}{\left(a\right)} - 3\right) \left(\operatorname{re}{\left(a\right)} + 4\right) + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 7 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 3\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
(im(a)^2 + (-3 + re(a))*(4 + re(a)) - 7*i*im(a))/((-3 + re(a))^2 + im(a)^2)