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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
(tan⁡2(x)−5tan⁡(x))−6=0\left(\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)}\right) - 6 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(tan⁡2(x)−5tan⁡(x))−6=0\left(\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)}\right) - 6 = 0
cambiamos
tan⁡2(x)−5tan⁡(x)−6=0\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)} - 6 = 0
(tan⁡2(x)−5tan⁡(x))−6=0\left(\tan^{2}{\left(x \right)} - 5 \tan{\left(x \right)}\right) - 6 = 0
Sustituimos
w=tan⁡(x)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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=−5b = -5
c=−6c = -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
w1=6w_{1} = 6
w2=−1w_{2} = -1
hacemos cambio inverso
tan⁡(x)=w\tan{\left(x \right)} = w
Tenemos la ecuación
tan⁡(x)=w\tan{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+atan⁡(w)x = \pi n + \operatorname{atan}{\left(w \right)}
O
x=πn+atan⁡(w)x = \pi n + \operatorname{atan}{\left(w \right)}
, donde n es cualquier número entero
sustituimos w:
x1=πn+atan⁡(w1)x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}
x1=πn+atan⁡(6)x_{1} = \pi n + \operatorname{atan}{\left(6 \right)}
x1=πn+atan⁡(6)x_{1} = \pi n + \operatorname{atan}{\left(6 \right)}
x2=πn+atan⁡(w2)x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}
x2=πn+atan⁡(−1)x_{2} = \pi n + \operatorname{atan}{\left(-1 \right)}
x2=πn−π4x_{2} = \pi n - \frac{\pi}{4}
Gráfica
0-80-60-40-2020406080-100100-1000000010000000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -pi 
x1 = ----
      4
x1=−π4x_{1} = - \frac{\pi}{4} 
x2 = atan(6)
x2=atan⁡(6)x_{2} = \operatorname{atan}{\left(6 \right)} 
x2 = atan(6)
Suma y producto de raíces [src] 
suma
  pi          
- -- + atan(6)
  4
−π4+atan⁡(6)- \frac{\pi}{4} + \operatorname{atan}{\left(6 \right)} 
=
  pi          
- -- + atan(6)
  4
−π4+atan⁡(6)- \frac{\pi}{4} + \operatorname{atan}{\left(6 \right)} 
producto
-pi         
----*atan(6)
 4
−π4atan⁡(6)- \frac{\pi}{4} \operatorname{atan}{\left(6 \right)} 
=
-pi*atan(6) 
------------
     4
−πatan⁡(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
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