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- La ecuación:
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Ecuación 5x+15=x-1
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Ecuación 3x^2=0
- Ecuación x^2-49=0
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Ecuación 1-x/2=x/3
- Expresar {x} en función de y en la ecuación:
- 4*x-1*y=-3
- -11*x+9*y=-20
- 9*x+17*y=13
- -19*x+14*y=20
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Expresiones idénticas
- (tgx+ctgy)^ dos -(tgx+ uno)(ctgy- uno)= cero
- (tgx más ctgy) al cuadrado menos (tgx más 1)(ctgy menos 1) es igual a 0
- (tgx más ctgy) en el grado dos menos (tgx más uno)(ctgy menos uno) es igual a cero
- (tgx+ctgy)2-(tgx+1)(ctgy-1)=0
- tgx+ctgy2-tgx+1ctgy-1=0
- (tgx+ctgy)²-(tgx+1)(ctgy-1)=0
- (tgx+ctgy) en el grado 2-(tgx+1)(ctgy-1)=0
- tgx+ctgy^2-tgx+1ctgy-1=0
- (tgx+ctgy)^2-(tgx+1)(ctgy-1)=O
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Expresiones semejantes
- (tgx+ctgy)^2-(tgx+1)(ctgy+1)=0
- (tgx+ctgy)^2-(tgx-1)(ctgy-1)=0
- (tgx-ctgy)^2-(tgx+1)(ctgy-1)=0
- (tgx+ctgy)^2+(tgx+1)(ctgy-1)=0
(tgx+ctgy)^2-(tgx+1)(ctgy-1)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
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2 (tan(x) + cot(y)) - (tan(x) + 1)*(cot(y) - 1) = 0
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} = 0$$
Solución detallada
Tenemos la ecuación
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} = 0$$
cambiamos
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} - 1 = 0$$
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} - 1 = 0$$
Sustituimos
$$w = \cot{\left(y \right)}$$
Abramos la expresión en la ecuación
$$- \left(w - 1\right) \left(\tan{\left(x \right)} + 1\right) + \left(w + \tan{\left(x \right)}\right)^{2} - 1 = 0$$
Obtenemos la ecuación cuadrática
$$w^{2} + w \tan{\left(x \right)} - w + \tan^{2}{\left(x \right)} + \tan{\left(x \right)} = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = \tan{\left(x \right)} - 1$$
$$c = \tan^{2}{\left(x \right)} + \tan{\left(x \right)}$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = \frac{\sqrt{\left(\tan{\left(x \right)} - 1\right)^{2} - 4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}}}{2} - \frac{\tan{\left(x \right)}}{2} + \frac{1}{2}$$
$$w_{2} = - \frac{\sqrt{\left(\tan{\left(x \right)} - 1\right)^{2} - 4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}}}{2} - \frac{\tan{\left(x \right)}}{2} + \frac{1}{2}$$
hacemos cambio inverso
$$\cot{\left(y \right)} = w$$
sustituimos w:
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} = 0$$
cambiamos
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} - 1 = 0$$
$$- \left(\tan{\left(x \right)} + 1\right) \left(\cot{\left(y \right)} - 1\right) + \left(\tan{\left(x \right)} + \cot{\left(y \right)}\right)^{2} - 1 = 0$$
Sustituimos
$$w = \cot{\left(y \right)}$$
Abramos la expresión en la ecuación
$$- \left(w - 1\right) \left(\tan{\left(x \right)} + 1\right) + \left(w + \tan{\left(x \right)}\right)^{2} - 1 = 0$$
Obtenemos la ecuación cuadrática
$$w^{2} + w \tan{\left(x \right)} - w + \tan^{2}{\left(x \right)} + \tan{\left(x \right)} = 0$$
Es la ecuación de la forma
a*w^2 + b*w + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = \tan{\left(x \right)} - 1$$
$$c = \tan^{2}{\left(x \right)} + \tan{\left(x \right)}$$
, entonces
D = b^2 - 4 * a * c =
(-1 + tan(x))^2 - 4 * (1) * (tan(x)^2 + tan(x)) = (-1 + tan(x))^2 - 4*tan(x)^2 - 4*tan(x)
La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{\left(\tan{\left(x \right)} - 1\right)^{2} - 4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}}}{2} - \frac{\tan{\left(x \right)}}{2} + \frac{1}{2}$$
$$w_{2} = - \frac{\sqrt{\left(\tan{\left(x \right)} - 1\right)^{2} - 4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)}}}{2} - \frac{\tan{\left(x \right)}}{2} + \frac{1}{2}$$
hacemos cambio inverso
$$\cot{\left(y \right)} = w$$
sustituimos w:
Gráfica
Respuesta rápida
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/ / ___ \\ / / ___ \\
| |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||
x1 = - re|atan|- + ------ - ---------------------|| - I*im|atan|- + ------ - ---------------------||
\ \2 2 2 // \ \2 2 2 //
$$x_{1} = - \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}$$
/ / ___ \\ / / ___ \\
| |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||
x2 = - re|atan|- + ------ + ---------------------|| - I*im|atan|- + ------ + ---------------------||
\ \2 2 2 // \ \2 2 2 //
$$x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}$$
x2 = -re(atan(sqrt(3)*i*(cot(y) - 1)/2 + cot(y)/2 + 1/2)) - i*im(atan(sqrt(3)*i*(cot(y) - 1)/2 + cot(y)/2 + 1/2))
Suma y producto de raíces
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suma
/ / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\
| |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||
- re|atan|- + ------ - ---------------------|| - I*im|atan|- + ------ - ---------------------|| + - re|atan|- + ------ + ---------------------|| - I*im|atan|- + ------ + ---------------------||
\ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 //
$$\left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}\right)$$
=
/ / ___ \\ / / ___ \\ / / ___ \\ / / ___ \\
| |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||
- re|atan|- + ------ + ---------------------|| - re|atan|- + ------ - ---------------------|| - I*im|atan|- + ------ + ---------------------|| - I*im|atan|- + ------ - ---------------------||
\ \2 2 2 // \ \2 2 2 // \ \2 2 2 // \ \2 2 2 //
$$- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}$$
producto
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||| | | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||| |- re|atan|- + ------ - ---------------------|| - I*im|atan|- + ------ - ---------------------|||*|- re|atan|- + ------ + ---------------------|| - I*im|atan|- + ------ + ---------------------||| \ \ \2 2 2 // \ \2 2 2 /// \ \ \2 2 2 // \ \2 2 2 ///
$$\left(- \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}\right)$$
=
/ / / ___ \\ / / ___ \\\ / / / ___ \\ / / ___ \\\ | | |1 cot(y) I*\/ 3 *(1 - cot(y))|| | |1 cot(y) I*\/ 3 *(1 - cot(y))||| | | |1 cot(y) I*\/ 3 *(-1 + cot(y))|| | |1 cot(y) I*\/ 3 *(-1 + cot(y))||| |I*im|atan|- + ------ + --------------------|| + re|atan|- + ------ + --------------------|||*|I*im|atan|- + ------ + ---------------------|| + re|atan|- + ------ + ---------------------||| \ \ \2 2 2 // \ \2 2 2 /// \ \ \2 2 2 // \ \2 2 2 ///
$$\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(1 - \cot{\left(y \right)}\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(1 - \cot{\left(y \right)}\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{\sqrt{3} i \left(\cot{\left(y \right)} - 1\right)}{2} + \frac{\cot{\left(y \right)}}{2} + \frac{1}{2} \right)}\right)}\right)$$
(i*im(atan(1/2 + cot(y)/2 + i*sqrt(3)*(1 - cot(y))/2)) + re(atan(1/2 + cot(y)/2 + i*sqrt(3)*(1 - cot(y))/2)))*(i*im(atan(1/2 + cot(y)/2 + i*sqrt(3)*(-1 + cot(y))/2)) + re(atan(1/2 + cot(y)/2 + i*sqrt(3)*(-1 + cot(y))/2)))