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6x^2-7y^2-8x+9y-10=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 6*x - 7*y - 8*x + 9*y - 10 = 0
$$\left(9 y + \left(- 8 x + \left(6 x^{2} - 7 y^{2}\right)\right)\right) - 10 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 6$$
$$b = -8$$
$$c = - 7 y^{2} + 9 y - 10$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{168 y^{2} - 216 y + 304}}{12} + \frac{2}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{168 y^{2} - 216 y + 304}}{12}$$
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
$$a = 6$$
$$b = -8$$
$$c = - 7 y^{2} + 9 y - 10$$
, entonces
D = b^2 - 4 * a * c =
(-8)^2 - 4 * (6) * (-10 - 7*y^2 + 9*y) = 304 - 216*y + 168*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{168 y^{2} - 216 y + 304}}{12} + \frac{2}{3}$$
$$x_{2} = \frac{2}{3} - \frac{\sqrt{168 y^{2} - 216 y + 304}}{12}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(9 y + \left(- 8 x + \left(6 x^{2} - 7 y^{2}\right)\right)\right) - 10 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} - \frac{7 y^{2}}{6} + \frac{3 y}{2} - \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{7 y^{2}}{6} + \frac{3 y}{2} - \frac{5}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = - \frac{7 y^{2}}{6} + \frac{3 y}{2} - \frac{5}{3}$$
$$\left(9 y + \left(- 8 x + \left(6 x^{2} - 7 y^{2}\right)\right)\right) - 10 = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} - \frac{7 y^{2}}{6} + \frac{3 y}{2} - \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{7 y^{2}}{6} + \frac{3 y}{2} - \frac{5}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = - \frac{7 y^{2}}{6} + \frac{3 y}{2} - \frac{5}{3}$$
Gráfica
Respuesta rápida
[src]
__________________________________________________________________________ __________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/|
\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *cos|------------------------------------------------------------------------| I*\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *sin|------------------------------------------------------------------------|
2 \ 2 / \ 2 /
x1 = - - ------------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------
3 6 6
$$x_{1} = - \frac{i \sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{2}{3}$$
__________________________________________________________________________ __________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/|
\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *cos|------------------------------------------------------------------------| I*\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *sin|------------------------------------------------------------------------|
2 \ 2 / \ 2 /
x2 = - + ------------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------
3 6 6
$$x_{2} = \frac{i \sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{2}{3}$$
x2 = i*((84*re(y)*im(y) - 54*im(y))^2 + (42*re(y)^2 - 54*re(y) - 42*im(y)^2 + 76)^2)^(1/4)*sin(atan2(84*re(y)*im(y) - 54*im(y, 42*re(y)^2 - 54*re(y) - 42*im(y)^2 + 76)/2)/6 + ((84*re(y)*im(y) - 54*im(y))^2 + (42*re(y)^2 - 54*re(y) - 42*im(y)^2 + 76)^2)^(1/4)*cos(atan2(84*re(y)*im(y) - 54*im(y), 42*re(y)^2 - 54*re(y) - 42*im(y)^2 + 76)/2)/6 + 2/3)
Suma y producto de raíces
[src]
suma
__________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/|
\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *cos|------------------------------------------------------------------------| I*\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *sin|------------------------------------------------------------------------| \/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *cos|------------------------------------------------------------------------| I*\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *sin|------------------------------------------------------------------------|
2 \ 2 / \ 2 / 2 \ 2 / \ 2 /
- - ------------------------------------------------------------------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------------- + - + ------------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------
3 6 6 3 6 6
$$\left(- \frac{i \sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{2}{3}\right) + \left(\frac{i \sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{2}{3}\right)$$
=
4/3
$$\frac{4}{3}$$
producto
/ __________________________________________________________________________ __________________________________________________________________________ \ / __________________________________________________________________________ __________________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/|| | 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/| 4 / 2 / 2 2 \ |atan2\-54*im(y) + 84*im(y)*re(y), 76 - 54*re(y) - 42*im (y) + 42*re (y)/|| | \/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *cos|------------------------------------------------------------------------| I*\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *sin|------------------------------------------------------------------------|| | \/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *cos|------------------------------------------------------------------------| I*\/ (-54*im(y) + 84*im(y)*re(y)) + \76 - 54*re(y) - 42*im (y) + 42*re (y)/ *sin|------------------------------------------------------------------------|| |2 \ 2 / \ 2 /| |2 \ 2 / \ 2 /| |- - ------------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------|*|- + ------------------------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------| \3 6 6 / \3 6 6 /
$$\left(- \frac{i \sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{2}{3}\right) \left(\frac{i \sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)}\right)^{2} + \left(42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(84 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 54 \operatorname{im}{\left(y\right)},42 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 54 \operatorname{re}{\left(y\right)} - 42 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 76 \right)}}{2} \right)}}{6} + \frac{2}{3}\right)$$
=
2 2 5 7*re (y) 3*re(y) 7*im (y) 3*I*im(y) 7*I*im(y)*re(y) - - - -------- + ------- + -------- + --------- - --------------- 3 6 2 6 2 3
$$- \frac{7 \left(\operatorname{re}{\left(y\right)}\right)^{2}}{6} - \frac{7 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)}}{3} + \frac{3 \operatorname{re}{\left(y\right)}}{2} + \frac{7 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{6} + \frac{3 i \operatorname{im}{\left(y\right)}}{2} - \frac{5}{3}$$
-5/3 - 7*re(y)^2/6 + 3*re(y)/2 + 7*im(y)^2/6 + 3*i*im(y)/2 - 7*i*im(y)*re(y)/3