Sr Examen
x²+11x+q=0 la ecuación
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Solución
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
$$a = 1$$
$$b = 11$$
$$c = q$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{121 - 4 q}}{2} - \frac{11}{2}$$
$$x_{2} = - \frac{\sqrt{121 - 4 q}}{2} - \frac{11}{2}$$
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
$$a = 1$$
$$b = 11$$
$$c = q$$
, entonces
D = b^2 - 4 * a * c =
(11)^2 - 4 * (1) * (q) = 121 - 4*q
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{121 - 4 q}}{2} - \frac{11}{2}$$
$$x_{2} = - \frac{\sqrt{121 - 4 q}}{2} - \frac{11}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 11$$
$$q = \frac{c}{a}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -11$$
$$x_{1} x_{2} = q$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 11$$
$$q = \frac{c}{a}$$
True
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -11$$
$$x_{1} x_{2} = q$$
Gráfica
Suma y producto de raíces
[src]
suma
______________________________ ______________________________ ______________________________ ______________________________
4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\
\/ (121 - 4*re(q)) + 16*im (q) *cos|------------------------------| I*\/ (121 - 4*re(q)) + 16*im (q) *sin|------------------------------| \/ (121 - 4*re(q)) + 16*im (q) *cos|------------------------------| I*\/ (121 - 4*re(q)) + 16*im (q) *sin|------------------------------|
11 \ 2 / \ 2 / 11 \ 2 / \ 2 /
- -- - --------------------------------------------------------------------- - ----------------------------------------------------------------------- + - -- + --------------------------------------------------------------------- + -----------------------------------------------------------------------
2 2 2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{11}{2}\right) + \left(\frac{i \sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{11}{2}\right)$$
=
-11
$$-11$$
producto
/ ______________________________ ______________________________ \ / ______________________________ ______________________________ \ | 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\| | 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\| | \/ (121 - 4*re(q)) + 16*im (q) *cos|------------------------------| I*\/ (121 - 4*re(q)) + 16*im (q) *sin|------------------------------|| | \/ (121 - 4*re(q)) + 16*im (q) *cos|------------------------------| I*\/ (121 - 4*re(q)) + 16*im (q) *sin|------------------------------|| | 11 \ 2 / \ 2 /| | 11 \ 2 / \ 2 /| |- -- - --------------------------------------------------------------------- - -----------------------------------------------------------------------|*|- -- + --------------------------------------------------------------------- + -----------------------------------------------------------------------| \ 2 2 2 / \ 2 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{11}{2}\right) \left(\frac{i \sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{11}{2}\right)$$
=
I*im(q) + re(q)
$$\operatorname{re}{\left(q\right)} + i \operatorname{im}{\left(q\right)}$$
i*im(q) + re(q)
Respuesta rápida
[src]
______________________________ ______________________________
4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\
\/ (121 - 4*re(q)) + 16*im (q) *cos|------------------------------| I*\/ (121 - 4*re(q)) + 16*im (q) *sin|------------------------------|
11 \ 2 / \ 2 /
x1 = - -- - --------------------------------------------------------------------- - -----------------------------------------------------------------------
2 2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{11}{2}$$
______________________________ ______________________________
4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\ 4 / 2 2 /atan2(-4*im(q), 121 - 4*re(q))\
\/ (121 - 4*re(q)) + 16*im (q) *cos|------------------------------| I*\/ (121 - 4*re(q)) + 16*im (q) *sin|------------------------------|
11 \ 2 / \ 2 /
x2 = - -- + --------------------------------------------------------------------- + -----------------------------------------------------------------------
2 2 2
$$x_{2} = \frac{i \sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(121 - 4 \operatorname{re}{\left(q\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(q\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(q\right)},121 - 4 \operatorname{re}{\left(q\right)} \right)}}{2} \right)}}{2} - \frac{11}{2}$$
x2 = i*((121 - 4*re(q))^2 + 16*im(q)^2)^(1/4)*sin(atan2(-4*im(q, 121 - 4*re(q))/2)/2 + ((121 - 4*re(q))^2 + 16*im(q)^2)^(1/4)*cos(atan2(-4*im(q), 121 - 4*re(q))/2)/2 - 11/2)