Sr Examen

Otras calculadoras

x^2+кx-16=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src] 
 2               
x  + k*x - 16 = 0
$$\left(k x + x^{2}\right) - 16 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = k$$
$$c = -16$$
, entonces
D = b^2 - 4 * a * c = 

(k)^2 - 4 * (1) * (-16) = 64 + k^2

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

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

o
$$x_{1} = - \frac{k}{2} + \frac{\sqrt{k^{2} + 64}}{2}$$
$$x_{2} = - \frac{k}{2} - \frac{\sqrt{k^{2} + 64}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = k$$
$$q = \frac{c}{a}$$
$$q = -16$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - k$$
$$x_{1} x_{2} = -16$$
Gráfica
Respuesta rápida [src] 
                 /              ___________________________________________                                                \       ___________________________________________                                                
                 |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
                 |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||   4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/|
                 |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||   \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *cos|------------------------------------------|
       re(k)     |  im(k)                                                      \                    2                     /|                                                      \                    2                     /
x1 = - ----- + I*|- ----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
         2       \    2                                                    2                                               /                                                  2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(k\right)}}{2}$$ 
                 /              ___________________________________________                                                \       ___________________________________________                                                
                 |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
                 |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||   4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/|
                 |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||   \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *cos|------------------------------------------|
       re(k)     |  im(k)                                                      \                    2                     /|                                                      \                    2                     /
x2 = - ----- + I*|- ----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
         2       \    2                                                    2                                               /                                                  2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(k\right)}}{2}$$ 
x2 = i*(((re(k)^2 - im(k)^2 + 64)^2 + 4*re(k)^2*im(k)^2)^(1/4)*sin(atan2(2*re(k)*im(k, re(k)^2 - im(k)^2 + 64)/2)/2 - im(k)/2) + ((re(k)^2 - im(k)^2 + 64)^2 + 4*re(k)^2*im(k)^2)^(1/4)*cos(atan2(2*re(k)*im(k), re(k)^2 - im(k)^2 + 64)/2)/2 - re(k)/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(k)*re(k), 64 + re (k) - im (k)/||   4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/|               |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||   4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/|
            |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||   \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *cos|------------------------------------------|               |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||   \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *cos|------------------------------------------|
  re(k)     |  im(k)                                                      \                    2                     /|                                                      \                    2                     /     re(k)     |  im(k)                                                      \                    2                     /|                                                      \                    2                     /
- ----- + I*|- ----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + - ----- + I*|- ----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
    2       \    2                                                    2                                               /                                                  2                                                      2       \    2                                                    2                                               /                                                  2
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(k\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(k\right)}}{2}\right)$$ 
=
           /              ___________________________________________                                                \     /              ___________________________________________                                                \
           |             /                       2                       /     /                      2        2   \\|     |             /                       2                       /     /                      2        2   \\|
           |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||     |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||
           |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||     |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||
           |  im(k)                                                      \                    2                     /|     |  im(k)                                                      \                    2                     /|
-re(k) + I*|- ----- + -----------------------------------------------------------------------------------------------| + I*|- ----- - -----------------------------------------------------------------------------------------------|
           \    2                                                    2                                               /     \    2                                                    2                                               /
$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) - \operatorname{re}{\left(k\right)}$$ 
producto
/            /              ___________________________________________                                                \       ___________________________________________                                                \ /            /              ___________________________________________                                                \       ___________________________________________                                                \
|            |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\| |            |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\|
|            |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||   4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/|| |            |          4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||   4 /  /       2        2   \        2      2        |atan2\2*im(k)*re(k), 64 + re (k) - im (k)/||
|            |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||   \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *cos|------------------------------------------|| |            |          \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *sin|------------------------------------------||   \/   \64 + re (k) - im (k)/  + 4*im (k)*re (k) *cos|------------------------------------------||
|  re(k)     |  im(k)                                                      \                    2                     /|                                                      \                    2                     /| |  re(k)     |  im(k)                                                      \                    2                     /|                                                      \                    2                     /|
|- ----- + I*|- ----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|- ----- + I*|- ----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------|
\    2       \    2                                                    2                                               /                                                  2                                               / \    2       \    2                                                    2                                               /                                                  2                                               /
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(k\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(k\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(k\right)}\right)^{2} \left(\operatorname{im}{\left(k\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(k\right)} \operatorname{im}{\left(k\right)},\left(\operatorname{re}{\left(k\right)}\right)^{2} - \left(\operatorname{im}{\left(k\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(k\right)}}{2}\right)$$ 
=
-16
$$-16$$ 
-16
    © Sr Examen – Калькулятор