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Expresiones semejantes
- y=x*y-y^2-2
- y=x*y+y^2+2
y=x*y-y^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
$$y = \left(x y - y^{2}\right) + 2$$
en
$$y + \left(\left(- x y + y^{2}\right) - 2\right) = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 1 - x$$
$$c = -2$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{x}{2} + \frac{\sqrt{\left(1 - x\right)^{2} + 8}}{2} - \frac{1}{2}$$
$$y_{2} = \frac{x}{2} - \frac{\sqrt{\left(1 - x\right)^{2} + 8}}{2} - \frac{1}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$y = \left(x y - y^{2}\right) + 2$$
en
$$y + \left(\left(- x y + y^{2}\right) - 2\right) = 0$$
Es la ecuación de la forma
a*y^2 + b*y + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 1 - x$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c =
(1 - x)^2 - 4 * (1) * (-2) = 8 + (1 - x)^2
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{x}{2} + \frac{\sqrt{\left(1 - x\right)^{2} + 8}}{2} - \frac{1}{2}$$
$$y_{2} = \frac{x}{2} - \frac{\sqrt{\left(1 - x\right)^{2} + 8}}{2} - \frac{1}{2}$$
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 1 - x$$
$$q = \frac{c}{a}$$
$$q = -2$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x - 1$$
$$y_{1} y_{2} = -2$$
$$p y + q + y^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 1 - x$$
$$q = \frac{c}{a}$$
$$q = -2$$
Fórmulas de Cardano-Vieta
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x - 1$$
$$y_{1} y_{2} = -2$$
Gráfica
Respuesta rápida
[src]
/ ________________________________________________________________ \ ________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|
| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *cos|--------------------------------------------------------------|
1 re(x) |im(x) \ 2 /| \ 2 /
y1 = - - + ----- + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 / 2
$$y_{1} = i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{1}{2}$$
/ ________________________________________________________________ \ ________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|
| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *cos|--------------------------------------------------------------|
1 re(x) |im(x) \ 2 /| \ 2 /
y2 = - - + ----- + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 / 2
$$y_{2} = i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{1}{2}$$
y2 = i*(((2*re(x)*im(x) - 2*im(x))^2 + (re(x)^2 - 2*re(x) - im(x)^2 + 9)^2)^(1/4)*sin(atan2(2*re(x)*im(x) - 2*im(x, re(x)^2 - 2*re(x) - im(x)^2 + 9)/2)/2 + im(x)/2) + ((2*re(x)*im(x) - 2*im(x))^2 + (re(x)^2 - 2*re(x) - im(x)^2 + 9)^2)^(1/4)*cos(atan2(2*re(x)*im(x) - 2*im(x), re(x)^2 - 2*re(x) - im(x)^2 + 9)/2)/2 + re(x)/2 - 1/2)
Suma y producto de raíces
[src]
suma
/ ________________________________________________________________ \ ________________________________________________________________ / ________________________________________________________________ \ ________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/| | 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|
| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *cos|--------------------------------------------------------------| | \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *cos|--------------------------------------------------------------|
1 re(x) |im(x) \ 2 /| \ 2 / 1 re(x) |im(x) \ 2 /| \ 2 /
- - + ----- + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------------- + - - + ----- + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------
2 2 \ 2 2 / 2 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{1}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{1}{2}\right)$$
=
/ ________________________________________________________________ \ / ________________________________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| | 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/||
| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| | \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------||
|im(x) \ 2 /| |im(x) \ 2 /|
-1 + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| + re(x)
\ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \operatorname{re}{\left(x\right)} - 1$$
producto
/ / ________________________________________________________________ \ ________________________________________________________________ \ / / ________________________________________________________________ \ ________________________________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| | | 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| 4 / 2 / 2 2 \ |atan2\-2*im(x) + 2*im(x)*re(x), 9 + re (x) - im (x) - 2*re(x)/|| | | \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *cos|--------------------------------------------------------------|| | | \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *sin|--------------------------------------------------------------|| \/ (-2*im(x) + 2*im(x)*re(x)) + \9 + re (x) - im (x) - 2*re(x)/ *cos|--------------------------------------------------------------|| | 1 re(x) |im(x) \ 2 /| \ 2 /| | 1 re(x) |im(x) \ 2 /| \ 2 /| |- - + ----- + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------|*|- - + ----- + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 \ 2 2 / 2 / \ 2 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{1}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - 2 \operatorname{im}{\left(x\right)},\left(\operatorname{re}{\left(x\right)}\right)^{2} - 2 \operatorname{re}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2} + 9 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{1}{2}\right)$$
=
-2
$$-2$$
-2