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a*x^2=c la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
   2    
a*x  = c
$$a x^{2} = c$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
$$a x^{2} = c$$
en
$$a x^{2} - c = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
True

$$b = 0$$
$$c = - c$$
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (a) * (-c) = 4*a*c

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

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

o
$$x_{1} = \frac{\sqrt{a c}}{a}$$
$$x_{2} = - \frac{\sqrt{a c}}{a}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$a x^{2} = c$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - c}{a} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{c}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{c}{a}$$
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
$$a x^{2} = c$$
Коэффициент при x равен
$$a$$
entonces son posibles los casos para a :
$$a < 0$$
$$a = 0$$
Consideremos todos los casos con detalles:
Con
$$a < 0$$
la ecuación será
$$- c - x^{2} = 0$$
su solución
$$x = - \sqrt{- c}$$
$$x = \sqrt{- c}$$
Con
$$a = 0$$
la ecuación será
$$- c = 0$$
su solución
Gráfica
Respuesta rápida [src]
                                /     /  /c\    /c\\\                              /     /  /c\    /c\\\
           _________________    |atan2|im|-|, re|-|||         _________________    |atan2|im|-|, re|-|||
          /   2/c\     2/c\     |     \  \a/    \a//|        /   2/c\     2/c\     |     \  \a/    \a//|
x1 = - 4 /  im |-| + re |-| *cos|-------------------| - I*4 /  im |-| + re |-| *sin|-------------------|
       \/      \a/      \a/     \         2         /     \/      \a/      \a/     \         2         /
$$x_{1} = - i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}$$
                              /     /  /c\    /c\\\                              /     /  /c\    /c\\\
         _________________    |atan2|im|-|, re|-|||         _________________    |atan2|im|-|, re|-|||
        /   2/c\     2/c\     |     \  \a/    \a//|        /   2/c\     2/c\     |     \  \a/    \a//|
x2 = 4 /  im |-| + re |-| *cos|-------------------| + I*4 /  im |-| + re |-| *sin|-------------------|
     \/      \a/      \a/     \         2         /     \/      \a/      \a/     \         2         /
$$x_{2} = i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}$$
x2 = i*(re(c/a)^2 + im(c/a)^2)^(1/4)*sin(atan2(im(c/a, re(c/a))/2) + (re(c/a)^2 + im(c/a)^2)^(1/4)*cos(atan2(im(c/a), re(c/a))/2))
Suma y producto de raíces [src]
suma
                           /     /  /c\    /c\\\                              /     /  /c\    /c\\\                            /     /  /c\    /c\\\                              /     /  /c\    /c\\\
      _________________    |atan2|im|-|, re|-|||         _________________    |atan2|im|-|, re|-|||       _________________    |atan2|im|-|, re|-|||         _________________    |atan2|im|-|, re|-|||
     /   2/c\     2/c\     |     \  \a/    \a//|        /   2/c\     2/c\     |     \  \a/    \a//|      /   2/c\     2/c\     |     \  \a/    \a//|        /   2/c\     2/c\     |     \  \a/    \a//|
- 4 /  im |-| + re |-| *cos|-------------------| - I*4 /  im |-| + re |-| *sin|-------------------| + 4 /  im |-| + re |-| *cos|-------------------| + I*4 /  im |-| + re |-| *sin|-------------------|
  \/      \a/      \a/     \         2         /     \/      \a/      \a/     \         2         /   \/      \a/      \a/     \         2         /     \/      \a/      \a/     \         2         /
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right)$$
=
0
$$0$$
producto
/                           /     /  /c\    /c\\\                              /     /  /c\    /c\\\\ /                         /     /  /c\    /c\\\                              /     /  /c\    /c\\\\
|      _________________    |atan2|im|-|, re|-|||         _________________    |atan2|im|-|, re|-|||| |    _________________    |atan2|im|-|, re|-|||         _________________    |atan2|im|-|, re|-||||
|     /   2/c\     2/c\     |     \  \a/    \a//|        /   2/c\     2/c\     |     \  \a/    \a//|| |   /   2/c\     2/c\     |     \  \a/    \a//|        /   2/c\     2/c\     |     \  \a/    \a//||
|- 4 /  im |-| + re |-| *cos|-------------------| - I*4 /  im |-| + re |-| *sin|-------------------||*|4 /  im |-| + re |-| *cos|-------------------| + I*4 /  im |-| + re |-| *sin|-------------------||
\  \/      \a/      \a/     \         2         /     \/      \a/      \a/     \         2         // \\/      \a/      \a/     \         2         /     \/      \a/      \a/     \         2         //
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right)$$
=
                               /  /c\    /c\\
     _________________  I*atan2|im|-|, re|-||
    /   2/c\     2/c\          \  \a/    \a//
-  /  im |-| + re |-| *e                     
 \/      \a/      \a/                        
$$- \sqrt{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}$$
-sqrt(im(c/a)^2 + re(c/a)^2)*exp(i*atan2(im(c/a), re(c/a)))