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(2-3x-2x^2)(2sin(x)-sqrt3)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
/             2\ /             ___\    
\2 - 3*x - 2*x /*\2*sin(x) - \/ 3 / = 0
(−2x2+(2−3x))(2sin⁡(x)−3)=0\left(- 2 x^{2} + \left(2 - 3 x\right)\right) \left(2 \sin{\left(x \right)} - \sqrt{3}\right) = 0
Simplificar
Solución detallada
Tenemos la ecuación
(−2x2+(2−3x))(2sin⁡(x)−3)=0\left(- 2 x^{2} + \left(2 - 3 x\right)\right) \left(2 \sin{\left(x \right)} - \sqrt{3}\right) = 0
cambiamos
(2sin⁡(x)−3)(−2x2−3x+2)=0\left(2 \sin{\left(x \right)} - \sqrt{3}\right) \left(- 2 x^{2} - 3 x + 2\right) = 0
(−2x2+(2−3x))(2sin⁡(x)−3)=0\left(- 2 x^{2} + \left(2 - 3 x\right)\right) \left(2 \sin{\left(x \right)} - \sqrt{3}\right) = 0
Sustituimos
w=sin⁡(x)w = \sin{\left(x \right)}
Tenemos la ecuación:
(-sqrt(3) + 2*w)*(2 - 3*x - 2*x^2) = 0

Abrimos la expresión:
-2*sqrt(3) + 4*w - 6*w*x - 4*w*x^2 + 2*sqrt(3)*x^2 + 3*x*sqrt(3) = 0

Reducimos, obtenemos:
-2*sqrt(3) + 4*w - 6*w*x - 4*w*x^2 + 2*sqrt(3)*x^2 + 3*x*sqrt(3) = 0

Abrimos los paréntesis en el miembro izquierdo de la ecuación
-2*sqrt3 + 4*w - 6*w*x - 4*w*x^2 + 2*sqrt3x^2 + 3*x*sqrt3 = 0

Dividamos ambos miembros de la ecuación en (-2*sqrt(3) + 4*w - 6*w*x - 4*w*x^2 + 2*sqrt(3)*x^2 + 3*x*sqrt(3))/w
w = 0 / ((-2*sqrt(3) + 4*w - 6*w*x - 4*w*x^2 + 2*sqrt(3)*x^2 + 3*x*sqrt(3))/w)

Obtenemos la respuesta: w = sqrt(3)/2
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(32)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}
x1=2πn+π3x_{1} = 2 \pi n + \frac{\pi}{3}
x2=2πn−asin⁡(w1)+πx_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x2=2πn−asin⁡(32)+πx_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi
x2=2πn+2π3x_{2} = 2 \pi n + \frac{2 \pi}{3}
Gráfica
0-80-60-40-2020406080-100100-100000100000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = -2
x1=−2x_{1} = -2 
x2 = 1/2
x2=12x_{2} = \frac{1}{2} 
     pi
x3 = --
     3
x3=π3x_{3} = \frac{\pi}{3} 
     2*pi
x4 = ----
      3
x4=2π3x_{4} = \frac{2 \pi}{3} 
x4 = 2*pi/3
Suma y producto de raíces [src] 
suma
           pi   2*pi
-2 + 1/2 + -- + ----
           3     3
((−2+12)+π3)+2π3\left(\left(-2 + \frac{1}{2}\right) + \frac{\pi}{3}\right) + \frac{2 \pi}{3} 
=
-3/2 + pi
−32+π- \frac{3}{2} + \pi 
producto
-2  pi 2*pi
---*--*----
 2  3   3
2π3⋅−1π3\frac{2 \pi}{3} \cdot - 1 \frac{\pi}{3} 
=
     2
-2*pi 
------
  9
−2π29- \frac{2 \pi^{2}}{9} 
-2*pi^2/9
Respuesta numérica [src] 
x1 = -29.3215314335047
x2 = 45.0294947014537
x3 = -17.8023583703422
x4 = 96.342174710087
x5 = -80.634211442138
x6 = -92.1533845053006
x7 = -41.8879020478639
x8 = -35.6047167406843
x9 = -10.471975511966
x10 = -99.4837673636768
x11 = -93.2005820564972
x12 = -67.0206432765823
x13 = -2.0
x14 = 26.1799387799149
x15 = 33.5103216382911
x16 = 71.2094334813686
x17 = 82.7286065445312
x18 = 89.0117918517108
x19 = -60.7374579694027
x20 = -79.5870138909414
x21 = -48.1710873550435
x22 = -55.5014702134197
x23 = -36.6519142918809
x24 = -85.870199198121
x25 = 39.7935069454707
x26 = 76.4454212373516
x27 = 20.943951023932
x28 = 101.57816246607
x29 = 2.0943951023932
x30 = -5.23598775598299
x31 = -11.5191730631626
x32 = 57.5958653158129
x33 = 7.33038285837618
x34 = 64.9262481741891
x35 = -23.0383461263252
x36 = 32.4631240870945
x37 = 14.6607657167524
x38 = 83.7758040957278
x39 = 13.6135681655558
x40 = 8.37758040957278
x41 = -24.0855436775217
x42 = 63.8790506229925
x43 = -30.3687289847013
x44 = 19.8967534727354
x45 = -68.0678408277789
x46 = 90.0589894029074
x47 = 95.2949771588904
x48 = 27.2271363311115
x49 = -98.4365698124802
x50 = -49.2182849062401
x51 = 58.6430628670095
x52 = 77.4926187885482
x53 = 51.3126800086333
x54 = -73.3038285837618
x55 = 0.5
x56 = -61.7846555205993
x57 = -54.4542726622231
x58 = 46.0766922526503
x59 = 70.162235930172
x60 = 52.3598775598299
x61 = -4.18879020478639
x62 = 1.0471975511966
x63 = -16.7551608191456
x64 = -74.3510261349584
x65 = 38.7463093942741
x66 = -86.9173967493176
x67 = -42.9350995990605
x67 = -42.9350995990605
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