Sr Examen

Otras calculadoras

2/tg^2x+7/tgx+5=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src] 
   2        7           
------- + ------ + 5 = 0
   2      tan(x)        
tan (x)
(2tan⁡2(x)+7tan⁡(x))+5=0\left(\frac{2}{\tan^{2}{\left(x \right)}} + \frac{7}{\tan{\left(x \right)}}\right) + 5 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(2tan⁡2(x)+7tan⁡(x))+5=0\left(\frac{2}{\tan^{2}{\left(x \right)}} + \frac{7}{\tan{\left(x \right)}}\right) + 5 = 0
cambiamos
5+7tan⁡(x)+2tan⁡2(x)=05 + \frac{7}{\tan{\left(x \right)}} + \frac{2}{\tan^{2}{\left(x \right)}} = 0
(2tan⁡2(x)+7tan⁡(x))+5=0\left(\frac{2}{\tan^{2}{\left(x \right)}} + \frac{7}{\tan{\left(x \right)}}\right) + 5 = 0
Sustituimos
w=tan⁡(x)w = \tan{\left(x \right)}
Tenemos la ecuación:
5+7w+2w2=05 + \frac{7}{w} + \frac{2}{w^{2}} = 0
Multipliquemos las dos partes de la ecuación por los denominadores:
w^2
obtendremos:
w2(5+7w+2w2)=0w^{2} \left(5 + \frac{7}{w} + \frac{2}{w^{2}}\right) = 0
5w2+7w+2=05 w^{2} + 7 w + 2 = 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:
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=5a = 5
b=7b = 7
c=2c = 2
, entonces
D = b^2 - 4 * a * c = 

(7)^2 - 4 * (5) * (2) = 9

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

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

o
w1=−25w_{1} = - \frac{2}{5}
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⁡(−25)x_{1} = \pi n + \operatorname{atan}{\left(- \frac{2}{5} \right)}
x1=πn−atan⁡(25)x_{1} = \pi n - \operatorname{atan}{\left(\frac{2}{5} \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
0-80-60-40-2020406080-100100-500000500000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
  pi            
- -- - atan(2/5)
  4
−π4−atan⁡(25)- \frac{\pi}{4} - \operatorname{atan}{\left(\frac{2}{5} \right)} 
=
             pi
-atan(2/5) - --
             4
−π4−atan⁡(25)- \frac{\pi}{4} - \operatorname{atan}{\left(\frac{2}{5} \right)} 
producto
-pi              
----*(-atan(2/5))
 4
−π4(−atan⁡(25))- \frac{\pi}{4} \left(- \operatorname{atan}{\left(\frac{2}{5} \right)}\right) 
=
pi*atan(2/5)
------------
     4
πatan⁡(25)4\frac{\pi \operatorname{atan}{\left(\frac{2}{5} \right)}}{4} 
pi*atan(2/5)/4
Respuesta rápida [src] 
     -pi 
x1 = ----
      4
x1=−π4x_{1} = - \frac{\pi}{4} 
x2 = -atan(2/5)
x2=−atan⁡(25)x_{2} = - \operatorname{atan}{\left(\frac{2}{5} \right)} 
x2 = -atan(2/5)
Respuesta numérica [src] 
x1 = -66.7588438887831
x2 = -60.0707667953184
x3 = 2.35619449019234
x4 = 87.1791961371168
x5 = 5.49778714378214
x6 = 93.8672732305814
x7 = 99.7455667514759
x8 = 33.7721210260903
x9 = -54.1924732744239
x10 = 96.6039740978861
x11 = -32.2013246992954
x12 = -0.785398163397448
x13 = 27.4889357189107
x14 = -41.6261026600648
x15 = 62.0464549083984
x16 = 58.9048622548086
x17 = 21.2057504117311
x18 = 8.63937979737193
x19 = 65.1880475619882
x20 = 80.8960108299372
x21 = -19.6349540849362
x22 = 78.1593099626325
x23 = -85.6083998103219
x24 = -82.4668071567321
x25 = 24.3473430653209
x26 = -82.061915370447
x27 = -35.3429173528852
x28 = -51.0508806208341
x29 = -97.769878638396
x30 = 43.1968989868597
x31 = -16.0884696450613
x32 = 77.7544181763474
x33 = 46.3384916404494
x34 = -31.7964329130103
x35 = -38.0796182201899
x36 = -88.7499924639117
x37 = -95.0331777710912
x38 = -3.92699081698724
x39 = -44.7676953136546
x40 = -98.174770424681
x41 = -79.3252145031423
x42 = 93.4623814442964
x43 = 36.9137136796801
x44 = 74.6128255227576
x45 = -13.3517687777566
x46 = 18.0641577581413
x47 = 11.7809724509617
x48 = 55.7632696012188
x49 = -7.06858347057703
x50 = -63.6172512351933
x51 = 5.90267893006722
x52 = -47.9092879672443
x53 = -9.80528433788174
x54 = 27.8938275051958
x55 = 40.0553063332699
x56 = -57.3340659280137
x57 = -25.9181393921158
x58 = -10.2101761241668
x59 = -91.8915851175014
x60 = -73.0420291959627
x61 = 34.1770128123754
x62 = -69.9004365423729
x63 = -29.0597320457056
x64 = 90.3207887907066
x65 = 12.1858642372468
x66 = -60.4756585816035
x67 = -75.7787300632674
x68 = 71.4712328691678
x69 = 14.9225651045515
x70 = 71.8761246554529
x71 = -38.484510006475
x72 = 100.150458537761
x73 = 30.6305283725005
x74 = 49.4800842940392
x75 = -53.7875814881388
x76 = -76.1836218495525
x77 = 56.1681613875039
x78 = 49.8849760803243
x79 = 52.621676947629
x80 = -16.4933614313464
x81 = 68.329640215578
x82 = -22.776546738526
x83 = 84.037603483527
x83 = 84.037603483527
    © Sr Examen – Калькулятор