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(a+4)(5-a)-(a-l)(a+8) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
(a + 4)*(5 - a) - (a - l)*(a + 8) = 0
$$\left(5 - a\right) \left(a + 4\right) - \left(a + 8\right) \left(a - l\right) = 0$$
Simplificar
Solución detallada
Tenemos la ecuación:
(a+4)*(5-a)-(a-l)*(a+8) = 0

Abrimos la expresión:
20 + a - a^2 - (a - l)*(a + 8) = 0

20 + a - a^2 - a^2 - 8*a + 8*l + a*l = 0

Reducimos, obtenemos:
20 - 7*a - 2*a^2 + 8*l + a*l = 0

Sumamos los términos semejantes en el miembro izquierdo de la ecuación:
20 - 7*a - 2*a^2 + 8*l + a*l = 0

Transportamos los términos libres (sin l)
del miembro izquierdo al derecho, obtenemos:
$$- 2 a^{2} + a l - 7 a + 8 l = -20$$
Move the summands with the other variables
del miembro izquierdo al derecho, obtenemos:
$$\left(-2\right) a^{2} + a l + 8 l = 7 a - 20$$
Dividamos ambos miembros de la ecuación en (-2*a^2 + 8*l + a*l)/l
l = -20 + 7*a / ((-2*a^2 + 8*l + a*l)/l)

Obtenemos la respuesta: l = (-20 + 2*a^2 + 7*a)/(8 + a)
Gráfica
Suma y producto de raíces [src] 
suma
  /                                        /          2          2             \      \               /          2          2             \                                  
  |(8 + re(a))*(7*im(a) + 4*im(a)*re(a))   \-20 - 2*im (a) + 2*re (a) + 7*re(a)/*im(a)|   (8 + re(a))*\-20 - 2*im (a) + 2*re (a) + 7*re(a)/   (7*im(a) + 4*im(a)*re(a))*im(a)
I*|------------------------------------- - -------------------------------------------| + ------------------------------------------------- + -------------------------------
  |                   2     2                                    2     2              |                            2     2                                    2     2        
  \        (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)           /                 (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)
$$\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 8\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 8\right) \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$ 
=
  /                                        /          2          2             \      \               /          2          2             \                                  
  |(8 + re(a))*(7*im(a) + 4*im(a)*re(a))   \-20 - 2*im (a) + 2*re (a) + 7*re(a)/*im(a)|   (8 + re(a))*\-20 - 2*im (a) + 2*re (a) + 7*re(a)/   (7*im(a) + 4*im(a)*re(a))*im(a)
I*|------------------------------------- - -------------------------------------------| + ------------------------------------------------- + -------------------------------
  |                   2     2                                    2     2              |                            2     2                                    2     2        
  \        (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)           /                 (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)
$$\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 8\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 8\right) \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$ 
producto
  /                                        /          2          2             \      \               /          2          2             \                                  
  |(8 + re(a))*(7*im(a) + 4*im(a)*re(a))   \-20 - 2*im (a) + 2*re (a) + 7*re(a)/*im(a)|   (8 + re(a))*\-20 - 2*im (a) + 2*re (a) + 7*re(a)/   (7*im(a) + 4*im(a)*re(a))*im(a)
I*|------------------------------------- - -------------------------------------------| + ------------------------------------------------- + -------------------------------
  |                   2     2                                    2     2              |                            2     2                                    2     2        
  \        (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)           /                 (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)
$$\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 8\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 8\right) \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$ 
=
  2                                /          2          2             \     /                   2          2                               \      
im (a)*(7 + 4*re(a)) + (8 + re(a))*\-20 - 2*im (a) + 2*re (a) + 7*re(a)/ + I*\20 - 7*re(a) - 2*re (a) + 2*im (a) + (7 + 4*re(a))*(8 + re(a))/*im(a)
---------------------------------------------------------------------------------------------------------------------------------------------------
                                                                          2     2                                                                  
                                                               (8 + re(a))  + im (a)
$$\frac{\left(\operatorname{re}{\left(a\right)} + 8\right) \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right) + \left(4 \operatorname{re}{\left(a\right)} + 7\right) \left(\operatorname{im}{\left(a\right)}\right)^{2} + i \left(\left(\operatorname{re}{\left(a\right)} + 8\right) \left(4 \operatorname{re}{\left(a\right)} + 7\right) - 2 \left(\operatorname{re}{\left(a\right)}\right)^{2} - 7 \operatorname{re}{\left(a\right)} + 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$ 
(im(a)^2*(7 + 4*re(a)) + (8 + re(a))*(-20 - 2*im(a)^2 + 2*re(a)^2 + 7*re(a)) + i*(20 - 7*re(a) - 2*re(a)^2 + 2*im(a)^2 + (7 + 4*re(a))*(8 + re(a)))*im(a))/((8 + re(a))^2 + im(a)^2)
Respuesta rápida [src] 
       /                                        /          2          2             \      \               /          2          2             \                                  
       |(8 + re(a))*(7*im(a) + 4*im(a)*re(a))   \-20 - 2*im (a) + 2*re (a) + 7*re(a)/*im(a)|   (8 + re(a))*\-20 - 2*im (a) + 2*re (a) + 7*re(a)/   (7*im(a) + 4*im(a)*re(a))*im(a)
l1 = I*|------------------------------------- - -------------------------------------------| + ------------------------------------------------- + -------------------------------
       |                   2     2                                    2     2              |                            2     2                                    2     2        
       \        (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)           /                 (8 + re(a))  + im (a)                      (8 + re(a))  + im (a)
$$l_{1} = \frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(4 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 7 \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} + 8\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(a\right)} + 8\right) \left(2 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 7 \operatorname{re}{\left(a\right)} - 2 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 20\right)}{\left(\operatorname{re}{\left(a\right)} + 8\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$ 
l1 = (4*re(a)*im(a) + 7*im(a))*im(a)/((re(a) + 8)^2 + im(a)^2) + i*((4*re(a)*im(a) + 7*im(a))*(re(a) + 8)/((re(a) + 8)^2 + im(a)^2) - (2*re(a)^2 + 7*re(a) - 2*im(a)^2 - 20)*im(a)/((re(a) + 8)^2 + im(a)^2)) + (re(a) + 8)*(2*re(a)^2 + 7*re(a) - 2*im(a)^2 - 20)/((re(a) + 8)^2 + im(a)^2)
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