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-2cos^2x-7cosx-3=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
       2                      
- 2*cos (x) - 7*cos(x) - 3 = 0
$$\left(- 2 \cos^{2}{\left(x \right)} - 7 \cos{\left(x \right)}\right) - 3 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(- 2 \cos^{2}{\left(x \right)} - 7 \cos{\left(x \right)}\right) - 3 = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - 7 \cos{\left(x \right)} - 5 = 0$$
$$\left(- 2 \cos^{2}{\left(x \right)} - 7 \cos{\left(x \right)}\right) - 3 = 0$$
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
$$a = -2$$
$$b = -7$$
$$c = -3$$
, entonces
D = b^2 - 4 * a * c = 

(-7)^2 - 4 * (-2) * (-3) = 25

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

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

o
$$w_{1} = -3$$
$$w_{2} = - \frac{1}{2}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(-3 \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(-3 \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{2 \pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(-3 \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(-3 \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x_{4} = \pi n - \frac{\pi}{3}$$
Gráfica
Suma y producto de raíces [src]
suma
2*pi   4*pi                                                                        
---- + ---- + -re(acos(-3)) + 2*pi - I*im(acos(-3)) + I*im(acos(-3)) + re(acos(-3))
 3      3                                                                          
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) + \left(\left(\frac{2 \pi}{3} + \frac{4 \pi}{3}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right)\right)$$
=
4*pi
$$4 \pi$$
producto
2*pi 4*pi                                                                        
----*----*(-re(acos(-3)) + 2*pi - I*im(acos(-3)))*(I*im(acos(-3)) + re(acos(-3)))
 3    3                                                                          
$$\frac{2 \pi}{3} \frac{4 \pi}{3} \left(- \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right)$$
=
     2                                                                        
-8*pi *(I*im(acos(-3)) + re(acos(-3)))*(-2*pi + I*im(acos(-3)) + re(acos(-3)))
------------------------------------------------------------------------------
                                      9                                       
$$- \frac{8 \pi^{2} \left(\operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}\right)}{9}$$
-8*pi^2*(i*im(acos(-3)) + re(acos(-3)))*(-2*pi + i*im(acos(-3)) + re(acos(-3)))/9
Respuesta rápida [src]
     2*pi
x1 = ----
      3  
$$x_{1} = \frac{2 \pi}{3}$$
     4*pi
x2 = ----
      3  
$$x_{2} = \frac{4 \pi}{3}$$
x3 = -re(acos(-3)) + 2*pi - I*im(acos(-3))
$$x_{3} = - \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}$$
x4 = I*im(acos(-3)) + re(acos(-3))
$$x_{4} = \operatorname{re}{\left(\operatorname{acos}{\left(-3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(-3 \right)}\right)}$$
x4 = re(acos(-3)) + i*im(acos(-3))
Respuesta numérica [src]
x1 = 167.551608191456
x2 = -29.3215314335047
x3 = 79.5870138909414
x4 = 16.7551608191456
x5 = 67.0206432765823
x6 = -58.6430628670095
x7 = 27.2271363311115
x8 = -4.18879020478639
x9 = -85.870199198121
x10 = -67.0206432765823
x11 = 9818.5242400193
x12 = 54.4542726622231
x13 = 71.2094334813686
x14 = 52.3598775598299
x15 = -60.7374579694027
x16 = 10.471975511966
x17 = 83.7758040957278
x18 = 58.6430628670095
x19 = -73.3038285837618
x20 = -20.943951023932
x21 = 77.4926187885482
x22 = -79.5870138909414
x23 = -48.1710873550435
x24 = -8.37758040957278
x25 = 85.870199198121
x26 = -11992.5063563034
x27 = -33.5103216382911
x28 = 20.943951023932
x29 = -10.471975511966
x30 = 2.0943951023932
x31 = 96.342174710087
x32 = 33.5103216382911
x33 = -96.342174710087
x34 = -90.0589894029074
x35 = 60.7374579694027
x36 = -14.6607657167524
x37 = 39.7935069454707
x38 = -23.0383461263252
x39 = 4.18879020478639
x40 = 64.9262481741891
x41 = 90.0589894029074
x42 = 8.37758040957278
x43 = 46.0766922526503
x44 = -92.1533845053006
x45 = -2.0943951023932
x46 = 35.6047167406843
x47 = -362.330352714023
x48 = 98.4365698124802
x49 = 23.0383461263252
x50 = -46.0766922526503
x51 = -83.7758040957278
x52 = -52.3598775598299
x53 = 29.3215314335047
x54 = -27.2271363311115
x55 = 41.8879020478639
x56 = -54.4542726622231
x57 = -64.9262481741891
x58 = 92.1533845053006
x59 = -16.7551608191456
x60 = -39.7935069454707
x61 = 48.1710873550435
x62 = -35.6047167406843
x63 = 14.6607657167524
x64 = -41.8879020478639
x65 = -77.4926187885482
x66 = 73.3038285837618
x67 = -98.4365698124802
x68 = -71.2094334813686
x68 = -71.2094334813686