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Expresiones idénticas
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Expresiones semejantes
- log1+x(a-x+2)=2
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log1-x(a-x+2)=2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
log(1) - x*(a - x + 2) = 2
$$- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)} = 2$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)} = 2$$
en
$$\left(- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)}\right) - 2 = 0$$
Abramos la expresión en la ecuación
$$\left(- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)}\right) - 2 = 0$$
Obtenemos la ecuación cuadrática
$$- a x + x^{2} - 2 x - 2 = 0$$
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
$$a = 1$$
$$b = - a - 2$$
$$c = -2$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{a}{2} + \frac{\sqrt{\left(- a - 2\right)^{2} + 8}}{2} + 1$$
$$x_{2} = \frac{a}{2} - \frac{\sqrt{\left(- a - 2\right)^{2} + 8}}{2} + 1$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)} = 2$$
en
$$\left(- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)}\right) - 2 = 0$$
Abramos la expresión en la ecuación
$$\left(- x \left(\left(a - x\right) + 2\right) + \log{\left(1 \right)}\right) - 2 = 0$$
Obtenemos la ecuación cuadrática
$$- a x + x^{2} - 2 x - 2 = 0$$
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
$$a = 1$$
$$b = - a - 2$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c =
(-2 - a)^2 - 4 * (1) * (-2) = 8 + (-2 - a)^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{a}{2} + \frac{\sqrt{\left(- a - 2\right)^{2} + 8}}{2} + 1$$
$$x_{2} = \frac{a}{2} - \frac{\sqrt{\left(- a - 2\right)^{2} + 8}}{2} + 1$$
Gráfica
Respuesta rápida
[src]
/ ________________________________________________________________ \ ________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|
| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *cos|--------------------------------------------------------------|
re(a) |im(a) \ 2 /| \ 2 /
x1 = 1 + ----- + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} + 1$$
/ ________________________________________________________________ \ ________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|
| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *cos|--------------------------------------------------------------|
re(a) |im(a) \ 2 /| \ 2 /
x2 = 1 + ----- + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} + 1$$
x2 = i*(((2*re(a)*im(a) + 4*im(a))^2 + (re(a)^2 + 4*re(a) - im(a)^2 + 12)^2)^(1/4)*sin(atan2(2*re(a)*im(a) + 4*im(a, re(a)^2 + 4*re(a) - im(a)^2 + 12)/2)/2 + im(a)/2) + ((2*re(a)*im(a) + 4*im(a))^2 + (re(a)^2 + 4*re(a) - im(a)^2 + 12)^2)^(1/4)*cos(atan2(2*re(a)*im(a) + 4*im(a), re(a)^2 + 4*re(a) - im(a)^2 + 12)/2)/2 + re(a)/2 + 1)
Suma y producto de raíces
[src]
suma
/ ________________________________________________________________ \ ________________________________________________________________ / ________________________________________________________________ \ ________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/| | 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|
| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *cos|--------------------------------------------------------------| | \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *cos|--------------------------------------------------------------|
re(a) |im(a) \ 2 /| \ 2 / re(a) |im(a) \ 2 /| \ 2 /
1 + ----- + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------------------------------------------------------- + 1 + ----- + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} + 1\right) + \left(i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} + 1\right)$$
=
/ ________________________________________________________________ \ / ________________________________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| | 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/||
| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| | \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------||
|im(a) \ 2 /| |im(a) \ 2 /|
2 + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| + re(a)
\ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \operatorname{re}{\left(a\right)} + 2$$
producto
/ / ________________________________________________________________ \ ________________________________________________________________ \ / / ________________________________________________________________ \ ________________________________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| | | 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| 4 / 2 / 2 2 \ |atan2\4*im(a) + 2*im(a)*re(a), 12 + re (a) - im (a) + 4*re(a)/|| | | \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *cos|--------------------------------------------------------------|| | | \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *sin|--------------------------------------------------------------|| \/ (4*im(a) + 2*im(a)*re(a)) + \12 + re (a) - im (a) + 4*re(a)/ *cos|--------------------------------------------------------------|| | re(a) |im(a) \ 2 /| \ 2 /| | re(a) |im(a) \ 2 /| \ 2 /| |1 + ----- + I*|----- - ----------------------------------------------------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------------------------------------------------|*|1 + ----- + I*|----- + ----------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} + 1\right) \left(i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 12 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} + 1\right)$$
=
-2
$$-2$$
-2