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yx^2+y^2-2y-63+7x^2=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2    2                 2    
y*x  + y  - 2*y - 63 + 7*x  = 0
7x2+((−2y+(x2y+y2))−63)=07 x^{2} + \left(\left(- 2 y + \left(x^{2} y + y^{2}\right)\right) - 63\right) = 0
Simplificar
Solución detallada
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=y+7a = y + 7
b=0b = 0
c=y2−2y−63c = y^{2} - 2 y - 63
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (7 + y) * (-63 + y^2 - 2*y) = -(28 + 4*y)*(-63 + y^2 - 2*y)

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=−(4y+28)(y2−2y−63)2y+14x_{1} = \frac{\sqrt{- \left(4 y + 28\right) \left(y^{2} - 2 y - 63\right)}}{2 y + 14}
x2=−−(4y+28)(y2−2y−63)2y+14x_{2} = - \frac{\sqrt{- \left(4 y + 28\right) \left(y^{2} - 2 y - 63\right)}}{2 y + 14}
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
x2y+7x2+y2−2y−63=0x^{2} y + 7 x^{2} + y^{2} - 2 y - 63 = 0
Коэффициент при x равен
y+7y + 7
entonces son posibles los casos para y :
y<−7y < -7
y=−7y = -7
Consideremos todos los casos con detalles:
Con
y<−7y < -7
la ecuación será
17−x2=017 - x^{2} = 0
su solución
x=−17x = - \sqrt{17}
x=17x = \sqrt{17}
Con
y=−7y = -7
la ecuación será
0=00 = 0
su solución
cualquiera x
Teorema de Cardano-Vieta
reescribamos la ecuación
7x2+((−2y+(x2y+y2))−63)=07 x^{2} + \left(\left(- 2 y + \left(x^{2} y + y^{2}\right)\right) - 63\right) = 0
de
ax2+bx+c=0a x^{2} + b x + c = 0
como ecuación cuadrática reducida
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2y+7x2+y2−2y−63y+7=0\frac{x^{2} y + 7 x^{2} + y^{2} - 2 y - 63}{y + 7} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=y2−2y−63y+7q = \frac{y^{2} - 2 y - 63}{y + 7}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=0x_{1} + x_{2} = 0
x1x2=y2−2y−63y+7x_{1} x_{2} = \frac{y^{2} - 2 y - 63}{y + 7}
Gráfica
Suma y producto de raíces [src] 
suma
     _______________________                                      _______________________                                    _______________________                                      _______________________                              
  4 /            2     2        /atan2(-im(y), 9 - re(y))\     4 /            2     2        /atan2(-im(y), 9 - re(y))\   4 /            2     2        /atan2(-im(y), 9 - re(y))\     4 /            2     2        /atan2(-im(y), 9 - re(y))\
- \/  (9 - re(y))  + im (y) *cos|------------------------| - I*\/  (9 - re(y))  + im (y) *sin|------------------------| + \/  (9 - re(y))  + im (y) *cos|------------------------| + I*\/  (9 - re(y))  + im (y) *sin|------------------------|
                                \           2            /                                   \           2            /                                 \           2            /                                   \           2            /
(−i(9−re⁡(y))2+(im⁡(y))24sin⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)−(9−re⁡(y))2+(im⁡(y))24cos⁡(atan2⁡(−im⁡(y),9−re⁡(y))2))+(i(9−re⁡(y))2+(im⁡(y))24sin⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)+(9−re⁡(y))2+(im⁡(y))24cos⁡(atan2⁡(−im⁡(y),9−re⁡(y))2))\left(- i \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} - \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} + \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right) 
=
0
00 
producto
/     _______________________                                      _______________________                              \ /   _______________________                                      _______________________                              \
|  4 /            2     2        /atan2(-im(y), 9 - re(y))\     4 /            2     2        /atan2(-im(y), 9 - re(y))\| |4 /            2     2        /atan2(-im(y), 9 - re(y))\     4 /            2     2        /atan2(-im(y), 9 - re(y))\|
|- \/  (9 - re(y))  + im (y) *cos|------------------------| - I*\/  (9 - re(y))  + im (y) *sin|------------------------||*|\/  (9 - re(y))  + im (y) *cos|------------------------| + I*\/  (9 - re(y))  + im (y) *sin|------------------------||
\                                \           2            /                                   \           2            // \                              \           2            /                                   \           2            //
(−i(9−re⁡(y))2+(im⁡(y))24sin⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)−(9−re⁡(y))2+(im⁡(y))24cos⁡(atan2⁡(−im⁡(y),9−re⁡(y))2))(i(9−re⁡(y))2+(im⁡(y))24sin⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)+(9−re⁡(y))2+(im⁡(y))24cos⁡(atan2⁡(−im⁡(y),9−re⁡(y))2))\left(- i \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} - \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} + \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right) 
=
    ________________________                            
   /             2     2      I*atan2(-im(y), 9 - re(y))
-\/  (-9 + re(y))  + im (y) *e
−(re⁡(y)−9)2+(im⁡(y))2eiatan2⁡(−im⁡(y),9−re⁡(y))- \sqrt{\left(\operatorname{re}{\left(y\right)} - 9\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}} 
-sqrt((-9 + re(y))^2 + im(y)^2)*exp(i*atan2(-im(y), 9 - re(y)))
Respuesta rápida [src] 
          _______________________                                      _______________________                              
       4 /            2     2        /atan2(-im(y), 9 - re(y))\     4 /            2     2        /atan2(-im(y), 9 - re(y))\
x1 = - \/  (9 - re(y))  + im (y) *cos|------------------------| - I*\/  (9 - re(y))  + im (y) *sin|------------------------|
                                     \           2            /                                   \           2            /
x1=−i(9−re⁡(y))2+(im⁡(y))24sin⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)−(9−re⁡(y))2+(im⁡(y))24cos⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)x_{1} = - i \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} - \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} 
        _______________________                                      _______________________                              
     4 /            2     2        /atan2(-im(y), 9 - re(y))\     4 /            2     2        /atan2(-im(y), 9 - re(y))\
x2 = \/  (9 - re(y))  + im (y) *cos|------------------------| + I*\/  (9 - re(y))  + im (y) *sin|------------------------|
                                   \           2            /                                   \           2            /
x2=i(9−re⁡(y))2+(im⁡(y))24sin⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)+(9−re⁡(y))2+(im⁡(y))24cos⁡(atan2⁡(−im⁡(y),9−re⁡(y))2)x_{2} = i \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} + \sqrt[4]{\left(9 - \operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},9 - \operatorname{re}{\left(y\right)} \right)}}{2} \right)} 
x2 = i*((9 - re(y))^2 + im(y)^2)^(1/4)*sin(atan2(-im(y, 9 - re(y))/2) + ((9 - re(y))^2 + im(y)^2)^(1/4)*cos(atan2(-im(y), 9 - re(y))/2))
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