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- La ecuación:
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Ecuación (x+9)^2=36*x
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Ecuación a+120=2000/5
-
Ecuación (|x|)=6
-
Ecuación 6*x=12
- Expresar {x} en función de y en la ecuación:
- -6*x-12*y=9
- 9*x-1*y=17
- 10*x-2*y=11
- 12*x-3*y=-2
- Derivada de:
- f*(x)
-
Expresiones idénticas
- f*(x)= cuatro *x^ dos + cinco / dos *x^ dos + uno
- f multiplicar por (x) es igual a 4 multiplicar por x al cuadrado más 5 dividir por 2 multiplicar por x al cuadrado más 1
- f multiplicar por (x) es igual a cuatro multiplicar por x en el grado dos más cinco dividir por dos multiplicar por x en el grado dos más uno
- f*(x)=4*x2+5/2*x2+1
- f*x=4*x2+5/2*x2+1
- f*(x)=4*x²+5/2*x²+1
- f*(x)=4*x en el grado 2+5/2*x en el grado 2+1
- f(x)=4x^2+5/2x^2+1
- f(x)=4x2+5/2x2+1
- fx=4x2+5/2x2+1
- fx=4x^2+5/2x^2+1
- f*(x)=4*x^2+5 dividir por 2*x^2+1
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Expresiones semejantes
- f*(x)=4*x^2-5/2*x^2+1
- f*(x)=4*x^2+5/2*x^2-1
f*(x)=4*x^2+5/2*x^2+1 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2
2 5*x
f*x = 4*x + ---- + 1
2
$$f x = \left(\frac{5 x^{2}}{2} + 4 x^{2}\right) + 1$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$f x = \left(\frac{5 x^{2}}{2} + 4 x^{2}\right) + 1$$
en
$$f x + \left(\left(- 4 x^{2} - \frac{5 x^{2}}{2}\right) - 1\right) = 0$$
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
$$a = - \frac{13}{2}$$
$$b = f$$
$$c = -1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{f}{13} - \frac{\sqrt{f^{2} - 26}}{13}$$
$$x_{2} = \frac{f}{13} + \frac{\sqrt{f^{2} - 26}}{13}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$f x = \left(\frac{5 x^{2}}{2} + 4 x^{2}\right) + 1$$
en
$$f x + \left(\left(- 4 x^{2} - \frac{5 x^{2}}{2}\right) - 1\right) = 0$$
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
$$a = - \frac{13}{2}$$
$$b = f$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(f)^2 - 4 * (-13/2) * (-1) = -26 + f^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{f}{13} - \frac{\sqrt{f^{2} - 26}}{13}$$
$$x_{2} = \frac{f}{13} + \frac{\sqrt{f^{2} - 26}}{13}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$f x = \left(\frac{5 x^{2}}{2} + 4 x^{2}\right) + 1$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$- \frac{2 f x}{13} + x^{2} + \frac{2}{13} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{2 f}{13}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{13}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{2 f}{13}$$
$$x_{1} x_{2} = \frac{2}{13}$$
$$f x = \left(\frac{5 x^{2}}{2} + 4 x^{2}\right) + 1$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$- \frac{2 f x}{13} + x^{2} + \frac{2}{13} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{2 f}{13}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{13}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{2 f}{13}$$
$$x_{1} x_{2} = \frac{2}{13}$$
Gráfica
Respuesta rápida
[src]
/ ____________________________________________ \ ____________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|
| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|-------------------------------------------|
re(f) |im(f) \ 2 /| \ 2 /
x1 = ----- + I*|----- - -------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------
13 \ 13 13 / 13
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{re}{\left(f\right)}}{13}$$
/ ____________________________________________ \ ____________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|
| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|-------------------------------------------|
re(f) |im(f) \ 2 /| \ 2 /
x2 = ----- + I*|----- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------
13 \ 13 13 / 13
$$x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{re}{\left(f\right)}}{13}$$
x2 = i*(((re(f)^2 - im(f)^2 - 26)^2 + 4*re(f)^2*im(f)^2)^(1/4)*sin(atan2(2*re(f)*im(f, re(f)^2 - im(f)^2 - 26)/2)/13 + im(f)/13) + ((re(f)^2 - im(f)^2 - 26)^2 + 4*re(f)^2*im(f)^2)^(1/4)*cos(atan2(2*re(f)*im(f), re(f)^2 - im(f)^2 - 26)/2)/13 + re(f)/13)
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), -26 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/| | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|
| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|-------------------------------------------| | \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|-------------------------------------------|
re(f) |im(f) \ 2 /| \ 2 / re(f) |im(f) \ 2 /| \ 2 /
----- + I*|----- - -------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------- + ----- + I*|----- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------
13 \ 13 13 / 13 13 \ 13 13 / 13
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{re}{\left(f\right)}}{13}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{re}{\left(f\right)}}{13}\right)$$
=
/ ____________________________________________ \ / ____________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/||
| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| | \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------||
2*re(f) |im(f) \ 2 /| |im(f) \ 2 /|
------- + I*|----- - -------------------------------------------------------------------------------------------------| + I*|----- + -------------------------------------------------------------------------------------------------|
13 \ 13 13 / \ 13 13 /
$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\right)^{2} + 4 \left(\operatorname{re}{\left(f\right)}\right)^{2} \left(\operatorname{im}{\left(f\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(f\right)} \operatorname{im}{\left(f\right)},\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) + \frac{2 \operatorname{re}{\left(f\right)}}{13}$$
producto
/ / ____________________________________________ \ ____________________________________________ \ / / ____________________________________________ \ ____________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| | | 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(f)*re(f), -26 + re (f) - im (f)/|| | | \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|-------------------------------------------|| | | \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *sin|-------------------------------------------|| \/ \-26 + re (f) - im (f)/ + 4*im (f)*re (f) *cos|-------------------------------------------|| |re(f) |im(f) \ 2 /| \ 2 /| |re(f) |im(f) \ 2 /| \ 2 /| |----- + I*|----- - -------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------|*|----- + I*|----- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------| \ 13 \ 13 13 / 13 / \ 13 \ 13 13 / 13 /
$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{re}{\left(f\right)}}{13}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{im}{\left(f\right)}}{13}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} - \left(\operatorname{im}{\left(f\right)}\right)^{2} - 26\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} - 26 \right)}}{2} \right)}}{13} + \frac{\operatorname{re}{\left(f\right)}}{13}\right)$$
=
2/13
$$\frac{2}{13}$$
2/13