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sqrt(xy+x^2) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   __________    
  /        2     
\/  x*y + x   = 0
x2+xy=0\sqrt{x^{2} + x y} = 0
Simplificar
Solución detallada
x2+xy=0\sqrt{x^{2} + x y} = 0
cambiamos
x2+xy=0x^{2} + x y = 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:
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=1a = 1
b=yb = y
c=0c = 0
, entonces
D = b^2 - 4 * a * c = 

(y)^2 - 4 * (1) * (0) = y^2

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

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

o
x1=−y2+y22x_{1} = - \frac{y}{2} + \frac{\sqrt{y^{2}}}{2}
x2=−y2−y22x_{2} = - \frac{y}{2} - \frac{\sqrt{y^{2}}}{2}
Gráfica
Respuesta rápida [src] 
                 /              ______________________________________                                           \       ______________________________________                                           
                 |             /                  2                       /     /                 2        2   \\|      /                  2                       /     /                 2        2   \\
                 |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||   4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/|
                 |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||   \/   \re (y) - im (y)/  + 4*im (y)*re (y) *cos|-------------------------------------|
       re(y)     |  im(y)                                                 \                  2                  /|                                                 \                  2                  /
x1 = - ----- + I*|- ----- - -------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------
         2       \    2                                               2                                          /                                             2
x1=i(−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−re⁡(y)2x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(y\right)}}{2} 
                 /              ______________________________________                                           \       ______________________________________                                           
                 |             /                  2                       /     /                 2        2   \\|      /                  2                       /     /                 2        2   \\
                 |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||   4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/|
                 |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||   \/   \re (y) - im (y)/  + 4*im (y)*re (y) *cos|-------------------------------------|
       re(y)     |  im(y)                                                 \                  2                  /|                                                 \                  2                  /
x2 = - ----- + I*|- ----- + -------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------
         2       \    2                                               2                                          /                                             2
x2=i(((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)+((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−re⁡(y)2x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(y\right)}}{2} 
x2 = i*(((re(y)^2 - im(y)^2)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(2*re(y)*im(y, re(y)^2 - im(y)^2)/2)/2 - im(y)/2) + ((re(y)^2 - im(y)^2)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2)/2)/2 - re(y)/2)
Suma y producto de raíces [src] 
suma
            /              ______________________________________                                           \       ______________________________________                                                          /              ______________________________________                                           \       ______________________________________                                           
            |             /                  2                       /     /                 2        2   \\|      /                  2                       /     /                 2        2   \\               |             /                  2                       /     /                 2        2   \\|      /                  2                       /     /                 2        2   \\
            |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||   4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/|               |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||   4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/|
            |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||   \/   \re (y) - im (y)/  + 4*im (y)*re (y) *cos|-------------------------------------|               |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||   \/   \re (y) - im (y)/  + 4*im (y)*re (y) *cos|-------------------------------------|
  re(y)     |  im(y)                                                 \                  2                  /|                                                 \                  2                  /     re(y)     |  im(y)                                                 \                  2                  /|                                                 \                  2                  /
- ----- + I*|- ----- - -------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------- + - ----- + I*|- ----- + -------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------
    2       \    2                                               2                                          /                                             2                                                 2       \    2                                               2                                          /                                             2
(i(−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−re⁡(y)2)+(i(((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)+((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−re⁡(y)2)\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(y\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(y\right)}}{2}\right) 
=
           /              ______________________________________                                           \     /              ______________________________________                                           \
           |             /                  2                       /     /                 2        2   \\|     |             /                  2                       /     /                 2        2   \\|
           |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||     |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||
           |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||     |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||
           |  im(y)                                                 \                  2                  /|     |  im(y)                                                 \                  2                  /|
-re(y) + I*|- ----- + -------------------------------------------------------------------------------------| + I*|- ----- - -------------------------------------------------------------------------------------|
           \    2                                               2                                          /     \    2                                               2                                          /
i(−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)+i(((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)−re⁡(y)i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \operatorname{re}{\left(y\right)} 
producto
/            /              ______________________________________                                           \       ______________________________________                                           \ /            /              ______________________________________                                           \       ______________________________________                                           \
|            |             /                  2                       /     /                 2        2   \\|      /                  2                       /     /                 2        2   \\| |            |             /                  2                       /     /                 2        2   \\|      /                  2                       /     /                 2        2   \\|
|            |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||   4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/|| |            |          4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||   4 /  /  2        2   \        2      2        |atan2\2*im(y)*re(y), re (y) - im (y)/||
|            |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||   \/   \re (y) - im (y)/  + 4*im (y)*re (y) *cos|-------------------------------------|| |            |          \/   \re (y) - im (y)/  + 4*im (y)*re (y) *sin|-------------------------------------||   \/   \re (y) - im (y)/  + 4*im (y)*re (y) *cos|-------------------------------------||
|  re(y)     |  im(y)                                                 \                  2                  /|                                                 \                  2                  /| |  re(y)     |  im(y)                                                 \                  2                  /|                                                 \                  2                  /|
|- ----- + I*|- ----- - -------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------|*|- ----- + I*|- ----- + -------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------|
\    2       \    2                                               2                                          /                                             2                                          / \    2       \    2                                               2                                          /                                             2                                          /
(i(−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)−((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−re⁡(y)2)(i(((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−im⁡(y)2)+((re⁡(y))2−(im⁡(y))2)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(2re⁡(y)im⁡(y),(re⁡(y))2−(im⁡(y))2)2)2−re⁡(y)2)\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(y\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(y\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(y\right)}}{2}\right) 
=
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