Sr Examen
120x-12y=6x^2-6y^2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$120 x - 12 y = 6 x^{2} - 6 y^{2}$$
en
$$\left(120 x - 12 y\right) + \left(- 6 x^{2} + 6 y^{2}\right) = 0$$
Es la ecuación de la forma
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
$$a = -6$$
$$b = 120$$
$$c = 6 y^{2} - 12 y$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = 10 - \frac{\sqrt{144 y^{2} - 288 y + 14400}}{12}$$
$$x_{2} = \frac{\sqrt{144 y^{2} - 288 y + 14400}}{12} + 10$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$120 x - 12 y = 6 x^{2} - 6 y^{2}$$
en
$$\left(120 x - 12 y\right) + \left(- 6 x^{2} + 6 y^{2}\right) = 0$$
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
$$a = -6$$
$$b = 120$$
$$c = 6 y^{2} - 12 y$$
, entonces
D = b^2 - 4 * a * c =
(120)^2 - 4 * (-6) * (-12*y + 6*y^2) = 14400 - 288*y + 144*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = 10 - \frac{\sqrt{144 y^{2} - 288 y + 14400}}{12}$$
$$x_{2} = \frac{\sqrt{144 y^{2} - 288 y + 14400}}{12} + 10$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$120 x - 12 y = 6 x^{2} - 6 y^{2}$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 20 x - y^{2} + 2 y = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -20$$
$$q = \frac{c}{a}$$
$$q = - y^{2} + 2 y$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 20$$
$$x_{1} x_{2} = - y^{2} + 2 y$$
$$120 x - 12 y = 6 x^{2} - 6 y^{2}$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 20 x - y^{2} + 2 y = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -20$$
$$q = \frac{c}{a}$$
$$q = - y^{2} + 2 y$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 20$$
$$x_{1} x_{2} = - y^{2} + 2 y$$
Gráfica
Suma y producto de raíces
[src]
suma
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/|
10 - \/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *cos|----------------------------------------------------------------| - I*\/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *sin|----------------------------------------------------------------| + 10 + \/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *cos|----------------------------------------------------------------| + I*\/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + 10\right) + \left(i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + 10\right)$$
=
20
$$20$$
producto
/ __________________________________________________________________ __________________________________________________________________ \ / __________________________________________________________________ __________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/|| | 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/|| |10 - \/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *cos|----------------------------------------------------------------| - I*\/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *sin|----------------------------------------------------------------||*|10 + \/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *cos|----------------------------------------------------------------| + I*\/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *sin|----------------------------------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + 10\right) \left(i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + 10\right)$$
=
2 2 im (y) - re (y) + 2*re(y) + 2*I*im(y) - 2*I*im(y)*re(y)
$$- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + 2 \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2 i \operatorname{im}{\left(y\right)}$$
im(y)^2 - re(y)^2 + 2*re(y) + 2*i*im(y) - 2*i*im(y)*re(y)
Respuesta rápida
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/|
x1 = 10 - \/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *cos|----------------------------------------------------------------| - I*\/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} - \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + 10$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/| 4 / 2 / 2 2 \ |atan2\-2*im(y) + 2*im(y)*re(y), 100 + re (y) - im (y) - 2*re(y)/|
x2 = 10 + \/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *cos|----------------------------------------------------------------| + I*\/ (-2*im(y) + 2*im(y)*re(y)) + \100 + re (y) - im (y) - 2*re(y)/ *sin|----------------------------------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 2 \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 100 \right)}}{2} \right)} + 10$$
x2 = i*((2*re(y)*im(y) - 2*im(y))^2 + (re(y)^2 - 2*re(y) - im(y)^2 + 100)^2)^(1/4)*sin(atan2(2*re(y)*im(y) - 2*im(y, re(y)^2 - 2*re(y) - im(y)^2 + 100)/2) + ((2*re(y)*im(y) - 2*im(y))^2 + (re(y)^2 - 2*re(y) - im(y)^2 + 100)^2)^(1/4)*cos(atan2(2*re(y)*im(y) - 2*im(y), re(y)^2 - 2*re(y) - im(y)^2 + 100)/2) + 10)