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- Gráfico de la función y =:
- 2*cos(2)*(x-1)+4*x^2-8*x
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Expresiones semejantes
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Expresiones con funciones
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2*cos(2)*(x-1)+4*x^2-8*x=y la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2*cos(2)*(x - 1) + 4*x - 8*x = y
$$- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right) = y$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right) = y$$
en
$$- y + \left(- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right)\right) = 0$$
Abramos la expresión en la ecuación
$$- y + \left(- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right)\right) = 0$$
Obtenemos la ecuación cuadrática
$$4 x^{2} - 8 x + 2 x \cos{\left(2 \right)} - y - 2 \cos{\left(2 \right)} = 0$$
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
$$a = 4$$
$$b = -8 + 2 \cos{\left(2 \right)}$$
$$c = - y - 2 \cos{\left(2 \right)}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{16 y + 32 \cos{\left(2 \right)} + \left(-8 + 2 \cos{\left(2 \right)}\right)^{2}}}{8} - \frac{\cos{\left(2 \right)}}{4} + 1$$
$$x_{2} = - \frac{\sqrt{16 y + 32 \cos{\left(2 \right)} + \left(-8 + 2 \cos{\left(2 \right)}\right)^{2}}}{8} - \frac{\cos{\left(2 \right)}}{4} + 1$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right) = y$$
en
$$- y + \left(- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right)\right) = 0$$
Abramos la expresión en la ecuación
$$- y + \left(- 8 x + \left(4 x^{2} + \left(x - 1\right) 2 \cos{\left(2 \right)}\right)\right) = 0$$
Obtenemos la ecuación cuadrática
$$4 x^{2} - 8 x + 2 x \cos{\left(2 \right)} - y - 2 \cos{\left(2 \right)} = 0$$
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
$$a = 4$$
$$b = -8 + 2 \cos{\left(2 \right)}$$
$$c = - y - 2 \cos{\left(2 \right)}$$
, entonces
D = b^2 - 4 * a * c =
(-8 + 2*cos(2))^2 - 4 * (4) * (-y - 2*cos(2)) = (-8 + 2*cos(2))^2 + 16*y + 32*cos(2)
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{16 y + 32 \cos{\left(2 \right)} + \left(-8 + 2 \cos{\left(2 \right)}\right)^{2}}}{8} - \frac{\cos{\left(2 \right)}}{4} + 1$$
$$x_{2} = - \frac{\sqrt{16 y + 32 \cos{\left(2 \right)} + \left(-8 + 2 \cos{\left(2 \right)}\right)^{2}}}{8} - \frac{\cos{\left(2 \right)}}{4} + 1$$
Gráfica
Respuesta rápida
[src]
_______________________________________ _______________________________________
/ 2 / / 2 \\ / 2 / / 2 \\
4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/|
\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin|--------------------------------------|
cos(2) \ 2 / \ 2 /
x1 = 1 - ------ - --------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------
4 4 4
$$x_{1} = - \frac{i \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\cos{\left(2 \right)}}{4} + 1$$
_______________________________________ _______________________________________
/ 2 / / 2 \\ / 2 / / 2 \\
4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/|
\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin|--------------------------------------|
cos(2) \ 2 / \ 2 /
x2 = 1 - ------ + --------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------
4 4 4
$$x_{2} = \frac{i \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\cos{\left(2 \right)}}{4} + 1$$
x2 = i*((4*re(y) + cos(2)^2 + 16)^2 + 16*im(y)^2)^(1/4)*sin(atan2(4*im(y, 4*re(y) + cos(2)^2 + 16)/2)/4 + ((4*re(y) + cos(2)^2 + 16)^2 + 16*im(y)^2)^(1/4)*cos(atan2(4*im(y), 4*re(y) + cos(2)^2 + 16)/2)/4 - cos(2)/4 + 1)
Suma y producto de raíces
[src]
suma
_______________________________________ _______________________________________ _______________________________________ _______________________________________
/ 2 / / 2 \\ / 2 / / 2 \\ / 2 / / 2 \\ / 2 / / 2 \\
4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/|
\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin|--------------------------------------| \/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin|--------------------------------------|
cos(2) \ 2 / \ 2 / cos(2) \ 2 / \ 2 /
1 - ------ - --------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------- + 1 - ------ + --------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------
4 4 4 4 4 4
$$\left(- \frac{i \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\cos{\left(2 \right)}}{4} + 1\right) + \left(\frac{i \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\cos{\left(2 \right)}}{4} + 1\right)$$
=
cos(2)
2 - ------
2
$$2 - \frac{\cos{\left(2 \right)}}{2}$$
producto
/ _______________________________________ _______________________________________ \ / _______________________________________ _______________________________________ \ | / 2 / / 2 \\ / 2 / / 2 \\| | / 2 / / 2 \\ / 2 / / 2 \\| | 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/|| | 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| 4 / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/|| | \/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin|--------------------------------------|| | \/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin|--------------------------------------|| | cos(2) \ 2 / \ 2 /| | cos(2) \ 2 / \ 2 /| |1 - ------ - --------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------|*|1 - ------ + --------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------| \ 4 4 4 / \ 4 4 4 /
$$\left(- \frac{i \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\cos{\left(2 \right)}}{4} + 1\right) \left(\frac{i \sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{4} - \frac{\cos{\left(2 \right)}}{4} + 1\right)$$
=
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/ 2 / / 2 \\ / 2 / / 2 \\ / 2 / / 2 \\ / / 2 \\
/ / 2 \ 2 2|atan2\4*im(y), 16 + cos (2) + 4*re(y)/| / / 2 \ 2 2|atan2\4*im(y), 16 + cos (2) + 4*re(y)/| / / 2 \ 2 |atan2\4*im(y), 16 + cos (2) + 4*re(y)/| |atan2\4*im(y), 16 + cos (2) + 4*re(y)/|
2 \/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos |--------------------------------------| \/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *sin |--------------------------------------| I*\/ \16 + cos (2) + 4*re(y)/ + 16*im (y) *cos|--------------------------------------|*sin|--------------------------------------|
cos(2) cos (2) \ 2 / \ 2 / \ 2 / \ 2 /
1 - ------ + ------- - ---------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------
2 16 16 16 8
$$\frac{\sqrt{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{16} - \frac{i \sqrt{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{8} - \frac{\sqrt{\left(4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16\right)^{2} + 16 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos^{2}{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(y\right)},4 \operatorname{re}{\left(y\right)} + \cos^{2}{\left(2 \right)} + 16 \right)}}{2} \right)}}{16} + \frac{\cos^{2}{\left(2 \right)}}{16} - \frac{\cos{\left(2 \right)}}{2} + 1$$
1 - cos(2)/2 + cos(2)^2/16 - sqrt((16 + cos(2)^2 + 4*re(y))^2 + 16*im(y)^2)*cos(atan2(4*im(y), 16 + cos(2)^2 + 4*re(y))/2)^2/16 + sqrt((16 + cos(2)^2 + 4*re(y))^2 + 16*im(y)^2)*sin(atan2(4*im(y), 16 + cos(2)^2 + 4*re(y))/2)^2/16 - i*sqrt((16 + cos(2)^2 + 4*re(y))^2 + 16*im(y)^2)*cos(atan2(4*im(y), 16 + cos(2)^2 + 4*re(y))/2)*sin(atan2(4*im(y), 16 + cos(2)^2 + 4*re(y))/2)/8