Sr Examen
49p+14pq+q^2=196 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
$$q^{2} + \left(49 p + 14 p q\right) = 196$$
en
$$\left(q^{2} + \left(49 p + 14 p q\right)\right) - 196 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$q_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$q_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 14 p$$
$$c = 49 p - 196$$
, entonces
La ecuación tiene dos raíces.
o
$$q_{1} = - 7 p + \frac{\sqrt{196 p^{2} - 196 p + 784}}{2}$$
$$q_{2} = - 7 p - \frac{\sqrt{196 p^{2} - 196 p + 784}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$q^{2} + \left(49 p + 14 p q\right) = 196$$
en
$$\left(q^{2} + \left(49 p + 14 p q\right)\right) - 196 = 0$$
Es la ecuación de la forma
a*q^2 + b*q + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$q_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$q_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 14 p$$
$$c = 49 p - 196$$
, entonces
D = b^2 - 4 * a * c =
(14*p)^2 - 4 * (1) * (-196 + 49*p) = 784 - 196*p + 196*p^2
La ecuación tiene dos raíces.
q1 = (-b + sqrt(D)) / (2*a)
q2 = (-b - sqrt(D)) / (2*a)
o
$$q_{1} = - 7 p + \frac{\sqrt{196 p^{2} - 196 p + 784}}{2}$$
$$q_{2} = - 7 p - \frac{\sqrt{196 p^{2} - 196 p + 784}}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p q + q^{2} + q = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 14 p$$
$$q = \frac{c}{a}$$
$$q = 49 p - 196$$
Fórmulas de Cardano-Vieta
$$q_{1} + q_{2} = - p$$
$$q_{1} q_{2} = q$$
$$q_{1} + q_{2} = - 14 p$$
$$q_{1} q_{2} = 49 p - 196$$
$$p q + q^{2} + q = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 14 p$$
$$q = \frac{c}{a}$$
$$q = 49 p - 196$$
Fórmulas de Cardano-Vieta
$$q_{1} + q_{2} = - p$$
$$q_{1} q_{2} = q$$
$$q_{1} + q_{2} = - 14 p$$
$$q_{1} q_{2} = 49 p - 196$$
Gráfica
Respuesta rápida
[src]
/ ____________________________________________________________ \ ____________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|
q1 = -7*re(p) + I*|-7*im(p) - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *cos|----------------------------------------------------------|
\ \ 2 // \ 2 /
$$q_{1} = i \left(- 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) - 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{re}{\left(p\right)}$$
/ ____________________________________________________________ \ ____________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|
q2 = -7*re(p) + I*|-7*im(p) + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *cos|----------------------------------------------------------|
\ \ 2 // \ 2 /
$$q_{2} = i \left(7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) + 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{re}{\left(p\right)}$$
q2 = i*(7*((2*re(p)*im(p) - im(p))^2 + (re(p)^2 - re(p) - im(p)^2 + 4)^2)^(1/4)*sin(atan2(2*re(p)*im(p) - im(p, re(p)^2 - re(p) - im(p)^2 + 4)/2) - 7*im(p)) + 7*((2*re(p)*im(p) - im(p))^2 + (re(p)^2 - re(p) - im(p)^2 + 4)^2)^(1/4)*cos(atan2(2*re(p)*im(p) - im(p), re(p)^2 - re(p) - im(p)^2 + 4)/2) - 7*re(p))
Suma y producto de raíces
[src]
suma
/ ____________________________________________________________ \ ____________________________________________________________ / ____________________________________________________________ \ ____________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/| | 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|
-7*re(p) + I*|-7*im(p) - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *cos|----------------------------------------------------------| + -7*re(p) + I*|-7*im(p) + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *cos|----------------------------------------------------------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
$$\left(i \left(- 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) - 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{re}{\left(p\right)}\right) + \left(i \left(7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) + 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{re}{\left(p\right)}\right)$$
=
/ ____________________________________________________________ \ / ____________________________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| | 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/||
-14*re(p) + I*|-7*im(p) - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| + I*|-7*im(p) + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------||
\ \ 2 // \ \ 2 //
$$i \left(- 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) + i \left(7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) - 14 \operatorname{re}{\left(p\right)}$$
producto
/ / ____________________________________________________________ \ ____________________________________________________________ \ / / ____________________________________________________________ \ ____________________________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| | | 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| 4 / 2 / 2 2 \ |atan2\-im(p) + 2*im(p)*re(p), 4 + re (p) - im (p) - re(p)/|| |-7*re(p) + I*|-7*im(p) - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| - 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *cos|----------------------------------------------------------||*|-7*re(p) + I*|-7*im(p) + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *sin|----------------------------------------------------------|| + 7*\/ (-im(p) + 2*im(p)*re(p)) + \4 + re (p) - im (p) - re(p)/ *cos|----------------------------------------------------------|| \ \ \ 2 // \ 2 // \ \ \ 2 // \ 2 //
$$\left(i \left(- 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) - 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{re}{\left(p\right)}\right) \left(i \left(7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{im}{\left(p\right)}\right) + 7 \sqrt[4]{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)} - \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \operatorname{re}{\left(p\right)} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 4 \right)}}{2} \right)} - 7 \operatorname{re}{\left(p\right)}\right)$$
=
-196 + 49*re(p) + 49*I*im(p)
$$49 \operatorname{re}{\left(p\right)} + 49 i \operatorname{im}{\left(p\right)} - 196$$
-196 + 49*re(p) + 49*i*im(p)