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- La ecuación:
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Ecuación x=(x+1)/2
- Ecuación (x-6)(-5x-9)=0
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Ecuación x^2-14*x+33=0
-
Ecuación x^2+6=5x
- Expresar {x} en función de y en la ecuación:
- -13*x-19*y=17
- -9*x-12*y=-19
- -17*x-12*y=-9
- -8*x+3*y=-4
- Forma canónica:
- x+2y^2-4y+4
-
Expresiones idénticas
- x+ dos y^2- cuatro y+ cuatro = cero x-2y+4=0
- x más 2y al cuadrado menos 4y más 4 es igual a 0x menos 2y más 4 es igual a 0
- x más dos y al cuadrado menos cuatro y más cuatro es igual a cero x menos 2y más 4 es igual a 0
- x+2y2-4y+4=0x-2y+4=0
- x+2y²-4y+4=0x-2y+4=0
- x+2y en el grado 2-4y+4=0x-2y+4=0
- x+2y^2-4y+4=Ox-2y+4=O
-
Expresiones semejantes
- x+2y^2-4y+4=0x+2y+4=0
- x+2y^2-4y+4=0x-2y-4=0
- x+2y^2+4y+4=0x-2y+4=0
- x+2y^2-4y-4=0x-2y+4=0
- x-2y^2-4y+4=0x-2y+4=0
x+2y^2-4y+4=0x-2y+4=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 x + 2*y - 4*y + 4 = 0*x - 2*y + 4
$$\left(- 4 y + \left(x + 2 y^{2}\right)\right) + 4 = \left(0 x - 2 y\right) + 4$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 4 y + \left(x + 2 y^{2}\right)\right) + 4 = \left(0 x - 2 y\right) + 4$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{x}{2} + y^{2} - y = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = \frac{x}{2}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 1$$
$$y_{1} y_{2} = \frac{x}{2}$$
$$\left(- 4 y + \left(x + 2 y^{2}\right)\right) + 4 = \left(0 x - 2 y\right) + 4$$
de
$$a y^{2} + b y + c = 0$$
como ecuación cuadrática reducida
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{x}{2} + y^{2} - y = 0$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = \frac{x}{2}$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 1$$
$$y_{1} y_{2} = \frac{x}{2}$$
Gráfica
Respuesta rápida
[src]
___________________________ ___________________________
4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\
\/ (1 - 2*re(x)) + 4*im (x) *cos|----------------------------| I*\/ (1 - 2*re(x)) + 4*im (x) *sin|----------------------------|
1 \ 2 / \ 2 /
y1 = - - ---------------------------------------------------------------- - ------------------------------------------------------------------
2 2 2
$$y_{1} = - \frac{i \sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{1}{2}$$
___________________________ ___________________________
4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\
\/ (1 - 2*re(x)) + 4*im (x) *cos|----------------------------| I*\/ (1 - 2*re(x)) + 4*im (x) *sin|----------------------------|
1 \ 2 / \ 2 /
y2 = - + ---------------------------------------------------------------- + ------------------------------------------------------------------
2 2 2
$$y_{2} = \frac{i \sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{1}{2}$$
y2 = i*((1 - 2*re(x))^2 + 4*im(x)^2)^(1/4)*sin(atan2(-2*im(x, 1 - 2*re(x))/2)/2 + ((1 - 2*re(x))^2 + 4*im(x)^2)^(1/4)*cos(atan2(-2*im(x), 1 - 2*re(x))/2)/2 + 1/2)
Suma y producto de raíces
[src]
suma
___________________________ ___________________________ ___________________________ ___________________________
4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\
\/ (1 - 2*re(x)) + 4*im (x) *cos|----------------------------| I*\/ (1 - 2*re(x)) + 4*im (x) *sin|----------------------------| \/ (1 - 2*re(x)) + 4*im (x) *cos|----------------------------| I*\/ (1 - 2*re(x)) + 4*im (x) *sin|----------------------------|
1 \ 2 / \ 2 / 1 \ 2 / \ 2 /
- - ---------------------------------------------------------------- - ------------------------------------------------------------------ + - + ---------------------------------------------------------------- + ------------------------------------------------------------------
2 2 2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{1}{2}\right) + \left(\frac{i \sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{1}{2}\right)$$
=
1
$$1$$
producto
/ ___________________________ ___________________________ \ / ___________________________ ___________________________ \ | 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\| | 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\ 4 / 2 2 /atan2(-2*im(x), 1 - 2*re(x))\| | \/ (1 - 2*re(x)) + 4*im (x) *cos|----------------------------| I*\/ (1 - 2*re(x)) + 4*im (x) *sin|----------------------------|| | \/ (1 - 2*re(x)) + 4*im (x) *cos|----------------------------| I*\/ (1 - 2*re(x)) + 4*im (x) *sin|----------------------------|| |1 \ 2 / \ 2 /| |1 \ 2 / \ 2 /| |- - ---------------------------------------------------------------- - ------------------------------------------------------------------|*|- + ---------------------------------------------------------------- + ------------------------------------------------------------------| \2 2 2 / \2 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{1}{2}\right) \left(\frac{i \sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(1 - 2 \operatorname{re}{\left(x\right)}\right)^{2} + 4 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{im}{\left(x\right)},1 - 2 \operatorname{re}{\left(x\right)} \right)}}{2} \right)}}{2} + \frac{1}{2}\right)$$
=
re(x) I*im(x) ----- + ------- 2 2
$$\frac{\operatorname{re}{\left(x\right)}}{2} + \frac{i \operatorname{im}{\left(x\right)}}{2}$$
re(x)/2 + i*im(x)/2