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- La ecuación:
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Ecuación 7+x=4
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Ecuación 3-3*6+2=x
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Ecuación |x-2|+|x-4|=3
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Ecuación (x+2)^2=(1-x)^2
- Expresar {x} en función de y en la ecuación:
- 4*x+15*y=19
- 15*x+10*y=-4
- -2*x-14*y=7
- 12*x+14*y=-15
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Expresiones idénticas
- tg^ dos (x)+ctg^ dos (x)+ tres *(tg(x)+ctg(x))+ cuatro = cero
- tg al cuadrado (x) más ctg al cuadrado (x) más 3 multiplicar por (tg(x) más ctg(x)) más 4 es igual a 0
- tg en el grado dos (x) más ctg en el grado dos (x) más tres multiplicar por (tg(x) más ctg(x)) más cuatro es igual a cero
- tg2(x)+ctg2(x)+3*(tg(x)+ctg(x))+4=0
- tg2x+ctg2x+3*tgx+ctgx+4=0
- tg²(x)+ctg²(x)+3*(tg(x)+ctg(x))+4=0
- tg en el grado 2(x)+ctg en el grado 2(x)+3*(tg(x)+ctg(x))+4=0
- tg^2(x)+ctg^2(x)+3(tg(x)+ctg(x))+4=0
- tg2(x)+ctg2(x)+3(tg(x)+ctg(x))+4=0
- tg2x+ctg2x+3tgx+ctgx+4=0
- tg^2x+ctg^2x+3tgx+ctgx+4=0
- tg^2(x)+ctg^2(x)+3*(tg(x)+ctg(x))+4=O
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Expresiones semejantes
- tg^2(x)+ctg^2(x)+3*(tg(x)-ctg(x))+4=0
- tg^2(x)+ctg^2(x)-3*(tg(x)+ctg(x))+4=0
- tg^2(x)-ctg^2(x)+3*(tg(x)+ctg(x))+4=0
- tg^2(x)+ctg^2(x)+3*(tg(x)+ctg(x))-4=0
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Expresiones con funciones
- tg
- Tg\2asin\2a/tg\2a-sin\2a
- tgx\3=-1/2
- tg(x)=1
- tg(x/2)=0
- tgx*(2-cosx)=0
- ctg
- ctgx=1+2cos2x
- ctgx-x=0
- ctg(x)=0
- ctg(x)=1
- ctg(3*pi/2+2x)=-√3
- tg
- Tg\2asin\2a/tg\2a-sin\2a
- tgx\3=-1/2
- tg(x)=1
- tg(x/2)=0
- tgx*(2-cosx)=0
- ctg
- ctgx=1+2cos2x
- ctgx-x=0
- ctg(x)=0
- ctg(x)=1
- ctg(3*pi/2+2x)=-√3
tg^2(x)+ctg^2(x)+3*(tg(x)+ctg(x))+4=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 tan (x) + cot (x) + 3*(tan(x) + cot(x)) + 4 = 0
$$\left(3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right)\right) + 4 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right)\right) + 4 = 0$$
cambiamos
$$\tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 \cot{\left(x \right)} + 3 = 0$$
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 = 0$$
Sustituimos
$$w = \cot{\left(x \right)}$$
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
$$a = 1$$
$$b = 3$$
$$c = \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + 3$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
$$w_{2} = - \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
hacemos cambio inverso
$$\cot{\left(x \right)} = w$$
sustituimos w:
$$\left(3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right)\right) + 4 = 0$$
cambiamos
$$\tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 \cot{\left(x \right)} + 3 = 0$$
$$3 \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right) + \tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)} + 3 = 0$$
Sustituimos
$$w = \cot{\left(x \right)}$$
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
$$a = 1$$
$$b = 3$$
$$c = \tan^{2}{\left(x \right)} + 3 \tan{\left(x \right)} + 3$$
, entonces
D = b^2 - 4 * a * c =
(3)^2 - 4 * (1) * (3 + tan(x)^2 + 3*tan(x)) = -3 - 12*tan(x) - 4*tan(x)^2
La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
$$w_{2} = - \frac{\sqrt{- 4 \tan^{2}{\left(x \right)} - 12 \tan{\left(x \right)} - 3}}{2} - \frac{3}{2}$$
hacemos cambio inverso
$$\cot{\left(x \right)} = w$$
sustituimos w:
Suma y producto de raíces
[src]
suma
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ pi | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || - -- + - re|atan|- - -------|| - I*im|atan|- - -------|| + - re|atan|- + -------|| - I*im|atan|- + -------|| 4 \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) + \left(- \frac{\pi}{4} + \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 || pi | |1 I*\/ 3 || | |1 I*\/ 3 ||
- re|atan|- + -------|| - re|atan|- - -------|| - -- - I*im|atan|- + -------|| - I*im|atan|- - -------||
\ \2 2 // \ \2 2 // 4 \ \2 2 // \ \2 2 //
$$- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - \frac{\pi}{4} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
producto
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -pi | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| ----*|- re|atan|- - -------|| - I*im|atan|- - -------|||*|- re|atan|- + -------|| - I*im|atan|- + -------||| 4 \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$- \frac{\pi}{4} \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 |||
-pi*|I*im|atan|- + -------|| + re|atan|- + -------|||*|I*im|atan|- - -------|| + re|atan|- - -------|||
\ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
--------------------------------------------------------------------------------------------------------
4
$$- \frac{\pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)}{4}$$
-pi*(i*im(atan(1/2 + i*sqrt(3)/2)) + re(atan(1/2 + i*sqrt(3)/2)))*(i*im(atan(1/2 - i*sqrt(3)/2)) + re(atan(1/2 - i*sqrt(3)/2)))/4
Respuesta rápida
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-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x2 = - re|atan|- - -------|| - I*im|atan|- - -------||
\ \2 2 // \ \2 2 //
$$x_{2} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x3 = - re|atan|- + -------|| - I*im|atan|- + -------||
\ \2 2 // \ \2 2 //
$$x_{3} = - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
x3 = -re(atan(1/2 + sqrt(3)*i/2)) - i*im(atan(1/2 + sqrt(3)*i/2))
Respuesta numérica
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x1 = 40.0553062162188
x2 = -76.1836216304923
x3 = -54.1924731017967
x4 = -32.2013244485502
x5 = -98.1747706322284
x6 = 2.35619441168939
x7 = 30.6305285608463
x8 = 68.3296401581437
x9 = 18.0641575857363
x10 = -54.192473039953
x11 = -98.1747702202347
x12 = -63.6172513370425
x13 = 33.7721213315842
x14 = -41.6261027538328
x15 = 55.7632699272178
x16 = -91.8915846481544
x17 = 99.7455671224518
x18 = -3.92699077227342
x19 = -3.92699077276863
x20 = -25.9181399071463
x21 = 24.3473429940575
x22 = 46.3384915761999
x23 = -76.1836218812413
x24 = 14.9225649923453
x25 = 77.7544185241376
x26 = 90.3207887399165
x27 = -85.6083999205894
x28 = 62.0464548249302
x29 = 93.4623811044448
x30 = 96.6039735663169
x31 = -25.918139886171
x32 = -10.2101758562123
x33 = 84.0376034236404
x34 = -47.9092883924126
x35 = 11.7809727371379
x36 = -3.92699110792993
x37 = 74.6128249560098
x38 = -0.785398151967561
x39 = -19.6349541709315
x39 = -19.6349541709315