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tg²x-5tgx-6=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
   2                      
tan (x) - 5*tan(x) - 6 = 0
$$\left(\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)}\right) - 6 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)}\right) - 6 = 0$$
cambiamos
$$\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)} - 6 = 0$$
$$\left(\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)}\right) - 6 = 0$$
Sustituimos
$$w = \tan{\left(x \right)}$$
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
$$a = 1$$
$$b = -5$$
$$c = -6$$
, entonces
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (1) * (-6) = 49

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

o
$$w_{1} = 6$$
$$w_{2} = -1$$
hacemos cambio inverso
$$\tan{\left(x \right)} = w$$
Tenemos la ecuación
$$\tan{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
O
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(6 \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(6 \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(-1 \right)}$$
$$x_{2} = \pi n - \frac{\pi}{4}$$
Gráfica
Respuesta rápida [src]
     -pi 
x1 = ----
      4  
$$x_{1} = - \frac{\pi}{4}$$
x2 = atan(6)
$$x_{2} = \operatorname{atan}{\left(6 \right)}$$
x2 = atan(6)
Suma y producto de raíces [src]
suma
  pi          
- -- + atan(6)
  4           
$$- \frac{\pi}{4} + \operatorname{atan}{\left(6 \right)}$$
=
  pi          
- -- + atan(6)
  4           
$$- \frac{\pi}{4} + \operatorname{atan}{\left(6 \right)}$$
producto
-pi         
----*atan(6)
 4          
$$- \frac{\pi}{4} \operatorname{atan}{\left(6 \right)}$$
=
-pi*atan(6) 
------------
     4      
$$- \frac{\pi \operatorname{atan}{\left(6 \right)}}{4}$$
-pi*atan(6)/4
Respuesta numérica [src]
x1 = -66.7588438887831
x2 = 64.2375007211761
x3 = -23.7270935793381
x4 = 2.35619449019234
x5 = 87.1791961371168
x6 = 5.49778714378214
x7 = 99.7455667514759
x8 = 33.7721210260903
x9 = -54.1924732744239
x10 = 42.2463521460476
x11 = 96.6039740978861
x12 = -32.2013246992954
x13 = -0.785398163397448
x14 = 27.4889357189107
x15 = -41.6261026600648
x16 = 62.0464549083984
x17 = 58.9048622548086
x18 = 21.2057504117311
x19 = 8.63937979737193
x20 = 65.1880475619882
x21 = 80.8960108299372
x22 = -45.7182421544666
x23 = -19.6349540849362
x24 = -101.316363078271
x25 = -85.6083998103219
x26 = -82.4668071567321
x27 = 24.3473430653209
x28 = -35.3429173528852
x29 = -51.0508806208341
x30 = 13.9720182637394
x31 = 43.1968989868597
x32 = 77.7544181763474
x33 = -95.9837246119033
x34 = 46.3384916404494
x35 = -30.0102788865177
x36 = -88.7499924639117
x37 = -95.0331777710912
x38 = 79.9454639891251
x39 = -3.92699081698724
x40 = -44.7676953136546
x41 = -98.174770424681
x42 = 86.2286492963047
x43 = -79.3252145031423
x44 = -67.7093907295952
x45 = 20.255203570919
x46 = 93.4623814442964
x47 = 36.9137136796801
x48 = 74.6128255227576
x49 = -13.3517687777566
x50 = 18.0641577581413
x51 = 11.7809724509617
x52 = 57.9543154139965
x53 = 55.7632696012188
x54 = -7.06858347057703
x55 = -63.6172512351933
x56 = -47.9092879672443
x57 = -52.0014274616462
x58 = 40.0553063332699
x59 = -57.3340659280137
x60 = -25.9181393921158
x61 = -10.2101761241668
x62 = -91.8915851175014
x63 = -73.0420291959627
x64 = -1.73594500420952
x65 = -69.9004365423729
x66 = -29.0597320457056
x67 = 90.3207887907066
x68 = -73.9925760367748
x69 = -60.4756585816035
x70 = 71.4712328691678
x71 = 14.9225651045515
x72 = -38.484510006475
x73 = 30.6305283725005
x74 = -8.01913031138911
x75 = 49.4800842940392
x76 = 35.963166838868
x77 = -76.1836218495525
x78 = 52.621676947629
x79 = -16.4933614313464
x80 = 68.329640215578
x81 = -22.776546738526
x82 = 84.037603483527
x82 = 84.037603483527