Sr Examen

Otras calculadoras

4cos²x-4cosx+1=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                      
4*cos (x) - 4*cos(x) + 1 = 0
$$\left(4 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(4 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)}\right) + 1 = 0$$
cambiamos
$$\left(2 \cos{\left(x \right)} - 1\right)^{2} = 0$$
$$\left(4 \cos^{2}{\left(x \right)} - 4 \cos{\left(x \right)}\right) + 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
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
$$a = 4$$
$$b = -4$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c = 

(-4)^2 - 4 * (4) * (1) = 0

Como D = 0 hay sólo una raíz.
w = -b/2a = --4/2/(4)

$$w_{1} = \frac{1}{2}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{2} = \pi n - \frac{2 \pi}{3}$$
Gráfica
Suma y producto de raíces [src]
suma
pi   5*pi
-- + ----
3     3  
$$\frac{\pi}{3} + \frac{5 \pi}{3}$$
=
2*pi
$$2 \pi$$
producto
pi 5*pi
--*----
3   3  
$$\frac{\pi}{3} \frac{5 \pi}{3}$$
=
    2
5*pi 
-----
  9  
$$\frac{5 \pi^{2}}{9}$$
5*pi^2/9
Respuesta rápida [src]
     pi
x1 = --
     3 
$$x_{1} = \frac{\pi}{3}$$
     5*pi
x2 = ----
      3  
$$x_{2} = \frac{5 \pi}{3}$$
x2 = 5*pi/3
Respuesta numérica [src]
x1 = 13.6135684396201
x2 = 63.8790507271403
x3 = -45.0294951231126
x4 = 99.4837674682333
x5 = -70.1622357075444
x6 = 55.5014703257498
x7 = 70.1622360405453
x8 = -19.8967530598909
x9 = -61.7846554908967
x10 = -57.595865417166
x11 = 19.8967536749417
x12 = -26.1799386778372
x13 = 61.784655531313
x14 = -63.8790506566425
x15 = 1.04719806956205
x16 = 17.8023584488359
x17 = 38.7463092821203
x18 = -82.7286070571699
x19 = -5.23598786097547
x20 = -89.0117919593799
x21 = -80.6342110331773
x22 = 99.4837668784018
x23 = -74.351025882179
x24 = 68.0678405557106
x25 = 42.9350991937229
x26 = 82.7286064269614
x27 = -55.5014703215112
x28 = 38.7463098734355
x29 = -51.3126802812342
x30 = 17.8023584775534
x31 = -93.2005815528906
x32 = -49.2182844096022
x33 = 5.23598724822487
x34 = 49.218284401649
x35 = 55.5014697367214
x36 = 36.6519140400357
x37 = -11.5191730188737
x38 = -36.6519138745329
x39 = -89.0117922811961
x40 = 51.3126804421085
x41 = -76.4454216793275
x42 = -99.4837674758316
x43 = -7.33038312063536
x44 = 32.4631240681495
x45 = 93.2005815555239
x46 = 95.2949776000856
x47 = -63.8790506517573
x48 = 51.312680431024
x49 = -86.917396237751
x50 = 74.3510260669324
x51 = -30.368728721231
x52 = 86.9173963524527
x53 = -13.6135682558762
x54 = -55.5014701115869
x55 = 7.33038328389894
x56 = 32.463123988393
x57 = -68.0678407633572
x58 = 45.0294952225665
x59 = -38.7463099047238
x60 = 80.6342112007723
x61 = -11.5191731667677
x62 = 42.9350994138005
x63 = 76.4454211500423
x64 = -17.8023583209161
x65 = 61.7846556324545
x66 = 89.0117923751084
x67 = 26.1799388546927
x68 = -45.0294943159054
x69 = -70.1622358337434
x70 = 82.728607012954
x71 = 76.4454211439692
x72 = -24.0855435993091
x73 = -42.9350990838837
x74 = 11.5191725964414
x75 = -76.4454211107134
x76 = -49.2182850018891
x77 = -95.2949774417536
x78 = 57.5958656003702
x79 = 19.8967535641397
x80 = -93.200582140014
x81 = -1.04719729654138
x82 = -5.23598726745762
x83 = -32.4631245422958
x84 = -99.4837672100067
x85 = 11.5191731815621
x86 = -26.1799386297799
x87 = -1.04719796480249
x88 = -32.4631239640048
x89 = 30.3687289054827
x90 = 95.2949768473669
x90 = 95.2949768473669