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Expresiones idénticas
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a al cuadrado menos 32 multiplicar por x más 240 es igual a 0
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a2-32*x+240=0
a²-32*x+240=0
a en el grado 2-32*x+240=0
a^2-32x+240=0
a2-32x+240=0
a^2-32*x+240=O
Expresiones semejantes
a^2+32*x+240=0
a^2-32*x-240=0

a^2-32*x+240=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

 2                 
a  - 32*x + 240 = 0

$$\left(a^{2} - 32 x\right) + 240 = 0$$

Simplificar

Solución detallada

Es la ecuación de la forma

a*a^2 + b*a + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$a_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$a_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 0$$
$$c = 240 - 32 x$$
, entonces

D = b^2 - 4 * a * c =

(0)^2 - 4 * (1) * (240 - 32*x) = -960 + 128*x

La ecuación tiene dos raíces.

a1 = (-b + sqrt(D)) / (2*a)

a2 = (-b - sqrt(D)) / (2*a)

o
$$a_{1} = \frac{\sqrt{128 x - 960}}{2}$$
$$a_{2} = - \frac{\sqrt{128 x - 960}}{2}$$

Teorema de Cardano-Vieta

es ecuación cuadrática reducida
$$a^{2} + a p + q = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 240 - 32 x$$
Fórmulas de Cardano-Vieta
$$a_{1} + a_{2} = - p$$
$$a_{1} a_{2} = q$$
$$a_{1} + a_{2} = 0$$
$$a_{1} a_{2} = 240 - 32 x$$

Gráfica

Respuesta rápida [src]

            _____________________________                                             _____________________________                                   
         4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\
a1 = - 4*\/  (-15 + 2*re(x))  + 4*im (x) *cos|-----------------------------| - 4*I*\/  (-15 + 2*re(x))  + 4*im (x) *sin|-----------------------------|
                                             \              2              /                                           \              2              /

$$a_{1} = - 4 i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)} - 4 \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)}$$

          _____________________________                                             _____________________________                                   
       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\
a2 = 4*\/  (-15 + 2*re(x))  + 4*im (x) *cos|-----------------------------| + 4*I*\/  (-15 + 2*re(x))  + 4*im (x) *sin|-----------------------------|
                                           \              2              /                                           \              2              /

$$a_{2} = 4 i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)} + 4 \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)}$$

a2 = 4*i*((2*re(x) - 15)^2 + 4*im(x)^2)^(1/4)*sin(atan2(2*im(x, 2*re(x) - 15)/2) + 4*((2*re(x) - 15)^2 + 4*im(x)^2)^(1/4)*cos(atan2(2*im(x), 2*re(x) - 15)/2))

Suma y producto de raíces [src]

suma

       _____________________________                                             _____________________________                                           _____________________________                                             _____________________________                                   
    4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\     4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\
- 4*\/  (-15 + 2*re(x))  + 4*im (x) *cos|-----------------------------| - 4*I*\/  (-15 + 2*re(x))  + 4*im (x) *sin|-----------------------------| + 4*\/  (-15 + 2*re(x))  + 4*im (x) *cos|-----------------------------| + 4*I*\/  (-15 + 2*re(x))  + 4*im (x) *sin|-----------------------------|
                                        \              2              /                                           \              2              /                                         \              2              /                                           \              2              /

$$\left(- 4 i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)} - 4 \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)}\right) + \left(4 i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)} + 4 \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)}\right)$$

$$0$$

producto

/       _____________________________                                             _____________________________                                   \ /     _____________________________                                             _____________________________                                   \
|    4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\| |  4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\       4 /                2       2        /atan2(2*im(x), -15 + 2*re(x))\|
|- 4*\/  (-15 + 2*re(x))  + 4*im (x) *cos|-----------------------------| - 4*I*\/  (-15 + 2*re(x))  + 4*im (x) *sin|-----------------------------||*|4*\/  (-15 + 2*re(x))  + 4*im (x) *cos|-----------------------------| + 4*I*\/  (-15 + 2*re(x))  + 4*im (x) *sin|-----------------------------||
\                                        \              2              /                                           \              2              // \                                      \              2              /                                           \              2              //

$$\left(- 4 i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)} - 4 \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)}\right) \left(4 i \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)} + 4 \sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}{2} \right)}\right)$$

       _____________________________                                 
      /                2       2      I*atan2(2*im(x), -15 + 2*re(x))
-16*\/  (-15 + 2*re(x))  + 4*im (x) *e

$$- 16 \sqrt{\left(2 \operatorname{re}{\left(x\right)} - 15\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(x\right)},2 \operatorname{re}{\left(x\right)} - 15 \right)}}$$

-16*sqrt((-15 + 2*re(x))^2 + 4*im(x)^2)*exp(i*atan2(2*im(x), -15 + 2*re(x)))

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a^2-32*x+240=0 la ecuación

Solución numérica:

Solución