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- La ecuación:
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Ecuación 5*x=32+x
-
Ecuación 4*x-8=x+1
-
Ecuación 3*x^2+13*x-10=0
-
Ecuación 8-x^2=0
- Expresar {x} en función de y en la ecuación:
- -6*x+11*y=-5
- -15*x+7*y=1
- -11*x-11*y=4
- 11*x-9*y=-4
- Derivada de:
- f*(x)
-
Expresiones idénticas
- f*(x)= cuatro *x^ dos - dos
- f multiplicar por (x) es igual a 4 multiplicar por x al cuadrado menos 2
- f multiplicar por (x) es igual a cuatro multiplicar por x en el grado dos menos dos
- f*(x)=4*x2-2
- f*x=4*x2-2
- f*(x)=4*x²-2
- f*(x)=4*x en el grado 2-2
- f(x)=4x^2-2
- f(x)=4x2-2
- fx=4x2-2
- fx=4x^2-2
-
Expresiones semejantes
- f*x=x^7-4*x^5+2*x-1
- f*(x)=4*x^2+2
f*(x)=4*x^2-2 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
$$f x = 4 x^{2} - 2$$
en
$$f x + \left(2 - 4 x^{2}\right) = 0$$
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
$$a = -4$$
$$b = f$$
$$c = 2$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{f}{8} - \frac{\sqrt{f^{2} + 32}}{8}$$
$$x_{2} = \frac{f}{8} + \frac{\sqrt{f^{2} + 32}}{8}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$f x = 4 x^{2} - 2$$
en
$$f x + \left(2 - 4 x^{2}\right) = 0$$
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
$$a = -4$$
$$b = f$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(f)^2 - 4 * (-4) * (2) = 32 + f^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{f}{8} - \frac{\sqrt{f^{2} + 32}}{8}$$
$$x_{2} = \frac{f}{8} + \frac{\sqrt{f^{2} + 32}}{8}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$f x = 4 x^{2} - 2$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$- \frac{f x}{4} + x^{2} - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{f}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{f}{4}$$
$$x_{1} x_{2} = - \frac{1}{2}$$
$$f x = 4 x^{2} - 2$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$- \frac{f x}{4} + x^{2} - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{f}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{f}{4}$$
$$x_{1} x_{2} = - \frac{1}{2}$$
Gráfica
Respuesta rápida
[src]
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|
| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|------------------------------------------|
re(f) |im(f) \ 2 /| \ 2 /
x1 = ----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
8 \ 8 8 / 8
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{re}{\left(f\right)}}{8}$$
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|
| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|------------------------------------------|
re(f) |im(f) \ 2 /| \ 2 /
x2 = ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
8 \ 8 8 / 8
$$x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{re}{\left(f\right)}}{8}$$
x2 = i*(((re(f)^2 - im(f)^2 + 32)^2 + 4*re(f)^2*im(f)^2)^(1/4)*sin(atan2(2*re(f)*im(f, re(f)^2 - im(f)^2 + 32)/2)/8 + im(f)/8) + ((re(f)^2 - im(f)^2 + 32)^2 + 4*re(f)^2*im(f)^2)^(1/4)*cos(atan2(2*re(f)*im(f), re(f)^2 - im(f)^2 + 32)/2)/8 + re(f)/8)
Suma y producto de raíces
[src]
suma
/ ___________________________________________ \ ___________________________________________ / ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/| | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|
| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|------------------------------------------| | \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|------------------------------------------|
re(f) |im(f) \ 2 /| \ 2 / re(f) |im(f) \ 2 /| \ 2 /
----- + I*|----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + ----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
8 \ 8 8 / 8 8 \ 8 8 / 8
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{re}{\left(f\right)}}{8}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{re}{\left(f\right)}}{8}\right)$$
=
/ ___________________________________________ \ / ___________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/||
| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| | \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------||
re(f) |im(f) \ 2 /| |im(f) \ 2 /|
----- + I*|----- - -----------------------------------------------------------------------------------------------| + I*|----- + -----------------------------------------------------------------------------------------------|
4 \ 8 8 / \ 8 8 /
$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) + \frac{\operatorname{re}{\left(f\right)}}{4}$$
producto
/ / ___________________________________________ \ ___________________________________________ \ / / ___________________________________________ \ ___________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| | | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), 32 + re (f) - im (f)/|| | | \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|------------------------------------------|| | | \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|------------------------------------------|| \/ \32 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|------------------------------------------|| |re(f) |im(f) \ 2 /| \ 2 /| |re(f) |im(f) \ 2 /| \ 2 /| |----- + I*|----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|----- + I*|----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------| \ 8 \ 8 8 / 8 / \ 8 \ 8 8 / 8 /
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{re}{\left(f\right)}}{8}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{im}{\left(f\right)}}{8}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} + 32 \right)}}{2} \right)}}{8} + \frac{\operatorname{re}{\left(f\right)}}{8}\right)$$
=
-1/2
$$- \frac{1}{2}$$
-1/2