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x^2+2(a-3)x+9-2a=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 x + 2*(a - 3)*x + 9 - 2*a = 0
$$- 2 a + \left(\left(x^{2} + x 2 \left(a - 3\right)\right) + 9\right) = 0$$
Solución detallada
Abramos la expresión en la ecuación
$$- 2 a + \left(\left(x^{2} + x 2 \left(a - 3\right)\right) + 9\right) = 0$$
Obtenemos la ecuación cuadrática
$$2 a x - 2 a + x^{2} - 6 x + 9 = 0$$
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
$$a = 1$$
$$b = 2 a - 6$$
$$c = 9 - 2 a$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = - a + \frac{\sqrt{8 a + \left(2 a - 6\right)^{2} - 36}}{2} + 3$$
$$x_{2} = - a - \frac{\sqrt{8 a + \left(2 a - 6\right)^{2} - 36}}{2} + 3$$
$$- 2 a + \left(\left(x^{2} + x 2 \left(a - 3\right)\right) + 9\right) = 0$$
Obtenemos la ecuación cuadrática
$$2 a x - 2 a + x^{2} - 6 x + 9 = 0$$
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
$$a = 1$$
$$b = 2 a - 6$$
$$c = 9 - 2 a$$
, entonces
D = b^2 - 4 * a * c =
(-6 + 2*a)^2 - 4 * (1) * (9 - 2*a) = -36 + (-6 + 2*a)^2 + 8*a
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - a + \frac{\sqrt{8 a + \left(2 a - 6\right)^{2} - 36}}{2} + 3$$
$$x_{2} = - a - \frac{\sqrt{8 a + \left(2 a - 6\right)^{2} - 36}}{2} + 3$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2 a - 6$$
$$q = \frac{c}{a}$$
$$q = 9 - 2 a$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6 - 2 a$$
$$x_{1} x_{2} = 9 - 2 a$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2 a - 6$$
$$q = \frac{c}{a}$$
$$q = 9 - 2 a$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6 - 2 a$$
$$x_{1} x_{2} = 9 - 2 a$$
Gráfica
Respuesta rápida
[src]
/ ________________________________________________________________________ \ ________________________________________________________________________
| / 2 / / 2 \\| / 2 / / 2 \\
| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|
x1 = 3 - re(a) + I*|-im(a) - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *cos|----------------------------------------------------------------------|
\ \ 2 // \ 2 /
$$x_{1} = i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(a\right)} + 3$$
/ ________________________________________________________________________ \ ________________________________________________________________________
| / 2 / / 2 \\| / 2 / / 2 \\
| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|
x2 = 3 - re(a) + I*|-im(a) + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *cos|----------------------------------------------------------------------|
\ \ 2 // \ 2 /
$$x_{2} = i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) + \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(a\right)} + 3$$
x2 = i*((((re(a) - 4)*re(a) - im(a)^2)^2 + ((re(a) - 4)*im(a) + re(a)*im(a))^2)^(1/4)*sin(atan2((re(a) - 4)*im(a) + re(a)*im(a, (re(a) - 4)*re(a) - im(a)^2)/2) - im(a)) + (((re(a) - 4)*re(a) - im(a)^2)^2 + ((re(a) - 4)*im(a) + re(a)*im(a))^2)^(1/4)*cos(atan2((re(a) - 4)*im(a) + re(a)*im(a), (re(a) - 4)*re(a) - im(a)^2)/2) - re(a) + 3)
Suma y producto de raíces
[src]
suma
/ ________________________________________________________________________ \ ________________________________________________________________________ / ________________________________________________________________________ \ ________________________________________________________________________
| / 2 / / 2 \\| / 2 / / 2 \\ | / 2 / / 2 \\| / 2 / / 2 \\
| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/| | 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|
3 - re(a) + I*|-im(a) - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *cos|----------------------------------------------------------------------| + 3 - re(a) + I*|-im(a) + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *cos|----------------------------------------------------------------------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
$$\left(i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(a\right)} + 3\right) + \left(i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) + \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(a\right)} + 3\right)$$
=
/ ________________________________________________________________________ \ / ________________________________________________________________________ \
| / 2 / / 2 \\| | / 2 / / 2 \\|
| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| | 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/||
6 - 2*re(a) + I*|-im(a) + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| + I*|-im(a) - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------||
\ \ 2 // \ \ 2 //
$$i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) + i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) - 2 \operatorname{re}{\left(a\right)} + 6$$
producto
/ / ________________________________________________________________________ \ ________________________________________________________________________ \ / / ________________________________________________________________________ \ ________________________________________________________________________ \ | | / 2 / / 2 \\| / 2 / / 2 \\| | | / 2 / / 2 \\| / 2 / / 2 \\| | | 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| | | 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| 4 / / 2 \ 2 |atan2\(-4 + re(a))*im(a) + im(a)*re(a), - im (a) + (-4 + re(a))*re(a)/|| |3 - re(a) + I*|-im(a) - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| - \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *cos|----------------------------------------------------------------------||*|3 - re(a) + I*|-im(a) + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *sin|----------------------------------------------------------------------|| + \/ \- im (a) + (-4 + re(a))*re(a)/ + ((-4 + re(a))*im(a) + im(a)*re(a)) *cos|----------------------------------------------------------------------|| \ \ \ 2 // \ 2 // \ \ \ 2 // \ 2 //
$$\left(i \left(- \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) - \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(a\right)} + 3\right) \left(i \left(\sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(a\right)}\right) + \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{im}{\left(a\right)} + \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)} - 4\right) \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(a\right)} + 3\right)$$
=
9 - 2*re(a) - 2*I*im(a)
$$- 2 \operatorname{re}{\left(a\right)} - 2 i \operatorname{im}{\left(a\right)} + 9$$
9 - 2*re(a) - 2*i*im(a)