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(x^2+(1-a)*x-1)/x+3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2
x + (1 - a)*x - 1
------------------ + 3 = 0
x
$$3 + \frac{\left(x^{2} + x \left(1 - a\right)\right) - 1}{x} = 0$$
Solución detallada
Tenemos la ecuación:
$$3 + \frac{\left(x^{2} + x \left(1 - a\right)\right) - 1}{x} = 0$$
Multipliquemos las dos partes de la ecuación por los denominadores:
x
obtendremos:
$$x \left(3 + \frac{\left(x^{2} + x \left(1 - a\right)\right) - 1}{x}\right) = 0$$
$$- a x + x^{2} + 4 x - 1 = 0$$
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
$$a = 1$$
$$b = 4 - a$$
$$c = -1$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{a}{2} + \frac{\sqrt{\left(4 - a\right)^{2} + 4}}{2} - 2$$
$$x_{2} = \frac{a}{2} - \frac{\sqrt{\left(4 - a\right)^{2} + 4}}{2} - 2$$
$$3 + \frac{\left(x^{2} + x \left(1 - a\right)\right) - 1}{x} = 0$$
Multipliquemos las dos partes de la ecuación por los denominadores:
x
obtendremos:
$$x \left(3 + \frac{\left(x^{2} + x \left(1 - a\right)\right) - 1}{x}\right) = 0$$
$$- a x + x^{2} + 4 x - 1 = 0$$
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
$$a = 1$$
$$b = 4 - a$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(4 - a)^2 - 4 * (1) * (-1) = 4 + (4 - a)^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{a}{2} + \frac{\sqrt{\left(4 - a\right)^{2} + 4}}{2} - 2$$
$$x_{2} = \frac{a}{2} - \frac{\sqrt{\left(4 - a\right)^{2} + 4}}{2} - 2$$
Gráfica
Respuesta rápida
[src]
/ _________________________________________________________________ \ _________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|
| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *cos|---------------------------------------------------------------|
re(a) |im(a) \ 2 /| \ 2 /
x1 = -2 + ----- + I*|----- - ------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} - 2$$
/ _________________________________________________________________ \ _________________________________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|
| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *cos|---------------------------------------------------------------|
re(a) |im(a) \ 2 /| \ 2 /
x2 = -2 + ----- + I*|----- + ------------------------------------------------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} - 2$$
x2 = i*(((2*re(a)*im(a) - 8*im(a))^2 + (re(a)^2 - 8*re(a) - im(a)^2 + 20)^2)^(1/4)*sin(atan2(2*re(a)*im(a) - 8*im(a, re(a)^2 - 8*re(a) - im(a)^2 + 20)/2)/2 + im(a)/2) + ((2*re(a)*im(a) - 8*im(a))^2 + (re(a)^2 - 8*re(a) - im(a)^2 + 20)^2)^(1/4)*cos(atan2(2*re(a)*im(a) - 8*im(a), re(a)^2 - 8*re(a) - im(a)^2 + 20)/2)/2 + re(a)/2 - 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\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/| | 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|
| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *cos|---------------------------------------------------------------| | \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *cos|---------------------------------------------------------------|
re(a) |im(a) \ 2 /| \ 2 / re(a) |im(a) \ 2 /| \ 2 /
-2 + ----- + I*|----- - ------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------ + -2 + ----- + 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)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \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)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} - 2\right) + \left(i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \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)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} - 2\right)$$
=
/ _________________________________________________________________ \ / _________________________________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| | 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/||
| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| | \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------||
|im(a) \ 2 /| |im(a) \ 2 /|
-4 + I*|----- + ------------------------------------------------------------------------------------------------------------------------------------------| + I*|----- - ------------------------------------------------------------------------------------------------------------------------------------------| + re(a)
\ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \operatorname{re}{\left(a\right)} - 4$$
producto
/ / _________________________________________________________________ \ _________________________________________________________________ \ / / _________________________________________________________________ \ _________________________________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| | | 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| 4 / 2 / 2 2 \ |atan2\-8*im(a) + 2*im(a)*re(a), 20 + re (a) - im (a) - 8*re(a)/|| | | \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *cos|---------------------------------------------------------------|| | | \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *sin|---------------------------------------------------------------|| \/ (-8*im(a) + 2*im(a)*re(a)) + \20 + re (a) - im (a) - 8*re(a)/ *cos|---------------------------------------------------------------|| | re(a) |im(a) \ 2 /| \ 2 /| | re(a) |im(a) \ 2 /| \ 2 /| |-2 + ----- + I*|----- - ------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------|*|-2 + ----- + 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)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \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)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} - 2\right) \left(i \left(\frac{\sqrt[4]{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \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)} - 8 \operatorname{im}{\left(a\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 8 \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - 8 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} + \frac{\operatorname{re}{\left(a\right)}}{2} - 2\right)$$
=
-1
$$-1$$
-1