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5/2*x^2+y^2-6*x+2*y la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 5*x 2 ---- + y - 6*x + 2*y = 0 2
$$2 y + \left(- 6 x + \left(\frac{5 x^{2}}{2} + y^{2}\right)\right) = 0$$
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 2$$
$$c = \frac{5 x^{2}}{2} - 6 x$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{\sqrt{- 10 x^{2} + 24 x + 4}}{2} - 1$$
$$y_{2} = - \frac{\sqrt{- 10 x^{2} + 24 x + 4}}{2} - 1$$
a*y^2 + b*y + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 2$$
$$c = \frac{5 x^{2}}{2} - 6 x$$
, entonces
D = b^2 - 4 * a * c =
(2)^2 - 4 * (1) * (-6*x + 5*x^2/2) = 4 - 10*x^2 + 24*x
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{\sqrt{- 10 x^{2} + 24 x + 4}}{2} - 1$$
$$y_{2} = - \frac{\sqrt{- 10 x^{2} + 24 x + 4}}{2} - 1$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = \frac{5 x^{2}}{2} - 6 x$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = -2$$
$$y_{1} y_{2} = \frac{5 x^{2}}{2} - 6 x$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = \frac{5 x^{2}}{2} - 6 x$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = -2$$
$$y_{1} y_{2} = \frac{5 x^{2}}{2} - 6 x$$
Gráfica
Suma y producto de raíces
[src]
suma
________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/|
\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *cos|----------------------------------------------------------------------| I*\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *sin|----------------------------------------------------------------------| \/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *cos|----------------------------------------------------------------------| I*\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *sin|----------------------------------------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
-1 - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------- + -1 + -------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - 1\right) + \left(\frac{i \sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - 1\right)$$
=
-2
$$-2$$
producto
/ ________________________________________________________________________ ________________________________________________________________________ \ / ________________________________________________________________________ ________________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/|| | 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/|| | \/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *cos|----------------------------------------------------------------------| I*\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *sin|----------------------------------------------------------------------|| | \/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *cos|----------------------------------------------------------------------| I*\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *sin|----------------------------------------------------------------------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |-1 - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------|*|-1 + -------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 / \ 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - 1\right) \left(\frac{i \sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - 1\right)$$
=
2 2
5*im (x) 5*re (x)
-6*re(x) - -------- + -------- - 6*I*im(x) + 5*I*im(x)*re(x)
2 2
$$\frac{5 \left(\operatorname{re}{\left(x\right)}\right)^{2}}{2} + 5 i \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 6 \operatorname{re}{\left(x\right)} - \frac{5 \left(\operatorname{im}{\left(x\right)}\right)^{2}}{2} - 6 i \operatorname{im}{\left(x\right)}$$
-6*re(x) - 5*im(x)^2/2 + 5*re(x)^2/2 - 6*i*im(x) + 5*i*im(x)*re(x)
Respuesta rápida
[src]
________________________________________________________________________ ________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/|
\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *cos|----------------------------------------------------------------------| I*\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *sin|----------------------------------------------------------------------|
\ 2 / \ 2 /
y1 = -1 - -------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$y_{1} = - \frac{i \sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - 1$$
________________________________________________________________________ ________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/| 4 / 2 / 2 2 \ |atan2\24*im(x) - 20*im(x)*re(x), 4 - 10*re (x) + 10*im (x) + 24*re(x)/|
\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *cos|----------------------------------------------------------------------| I*\/ (24*im(x) - 20*im(x)*re(x)) + \4 - 10*re (x) + 10*im (x) + 24*re(x)/ *sin|----------------------------------------------------------------------|
\ 2 / \ 2 /
y2 = -1 + -------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$y_{2} = \frac{i \sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 20 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 24 \operatorname{im}{\left(x\right)},- 10 \left(\operatorname{re}{\left(x\right)}\right)^{2} + 24 \operatorname{re}{\left(x\right)} + 10 \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)}}{2} - 1$$
y2 = i*((-20*re(x)*im(x) + 24*im(x))^2 + (-10*re(x)^2 + 24*re(x) + 10*im(x)^2 + 4)^2)^(1/4)*sin(atan2(-20*re(x)*im(x) + 24*im(x, -10*re(x)^2 + 24*re(x) + 10*im(x)^2 + 4)/2)/2 + ((-20*re(x)*im(x) + 24*im(x))^2 + (-10*re(x)^2 + 24*re(x) + 10*im(x)^2 + 4)^2)^(1/4)*cos(atan2(-20*re(x)*im(x) + 24*im(x), -10*re(x)^2 + 24*re(x) + 10*im(x)^2 + 4)/2)/2 - 1)