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Ecuación x^2-6x+9=0
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Ecuación (x-11)^4=(x+3)^4
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Ecuación √(x^2-1)^3=2*x
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Ecuación -x/12+x=55/12
- Expresar {x} en función de y en la ecuación:
- 18*x-11*y=2
- 12*x-14*y=20
- -3*x-20*y=-13
- -16*x+14*y=-9
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Expresiones idénticas
- sin(x)*(dos *sin(x)- uno)-sqrt(tres)*sin(x)+sqrt(tres)/ dos = cero
- seno de (x) multiplicar por (2 multiplicar por seno de (x) menos 1) menos raíz cuadrada de (3) multiplicar por seno de (x) más raíz cuadrada de (3) dividir por 2 es igual a 0
- seno de (x) multiplicar por (dos multiplicar por seno de (x) menos uno) menos raíz cuadrada de (tres) multiplicar por seno de (x) más raíz cuadrada de (tres) dividir por dos es igual a cero
- sin(x)*(2*sin(x)-1)-√(3)*sin(x)+√(3)/2=0
- sin(x)(2sin(x)-1)-sqrt(3)sin(x)+sqrt(3)/2=0
- sinx2sinx-1-sqrt3sinx+sqrt3/2=0
- sin(x)*(2*sin(x)-1)-sqrt(3)*sin(x)+sqrt(3)/2=O
- sin(x)*(2*sin(x)-1)-sqrt(3)*sin(x)+sqrt(3) dividir por 2=0
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Expresiones semejantes
- sin(x)*(2*sin(x)-1)-sqrt(3)*sin(x)-sqrt(3)/2=0
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- sin(x)*(2*sin(x)-1)+sqrt(3)*sin(x)+sqrt(3)/2=0
- sinx*(2*sinx-1)-sqrt(3)*sinx+sqrt(3)/2=0
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Expresiones con funciones
- Seno sin
- sin(x)=-1/5
- sin2x=0
- sin(3*x-pi/6)=1/2
- sin(x)=-3/2
- sin(x)^3+cos(x)^3=1
- Seno sin
- sin(x)=-1/5
- sin2x=0
- sin(3*x-pi/6)=1/2
- sin(x)=-3/2
- sin(x)^3+cos(x)^3=1
- Raíz cuadrada sqrt
- sqrt(x^2-4)+sqrt(x^2+x-2)=0
- sqrt(x-2)=6
- sqrt(x-3)=5-x
- sqrt(x+4)=x-2
- sqrt(-72-17x)=-x
- Seno sin
- sin(x)=-1/5
- sin2x=0
- sin(3*x-pi/6)=1/2
- sin(x)=-3/2
- sin(x)^3+cos(x)^3=1
- Raíz cuadrada sqrt
- sqrt(x^2-4)+sqrt(x^2+x-2)=0
- sqrt(x-2)=6
- sqrt(x-3)=5-x
- sqrt(x+4)=x-2
- sqrt(-72-17x)=-x
sin(x)*(2*sin(x)-1)-sqrt(3)*sin(x)+sqrt(3)/2=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
___
___ \/ 3
sin(x)*(2*sin(x) - 1) - \/ 3 *sin(x) + ----- = 0
2
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - \sqrt{3} \sin{\left(x \right)} - \sin{\left(x \right)} + \frac{\sqrt{3}}{2} = 0$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Abramos la expresión en la ecuación
$$w \left(2 w - 1\right) - \sqrt{3} w + \frac{\sqrt{3}}{2} = 0$$
Obtenemos la ecuación cuadrática
$$2 w^{2} - \sqrt{3} w - w + \frac{\sqrt{3}}{2} = 0$$
Es la ecuación de la forma
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
$$a = 2$$
$$b = - \sqrt{3} - 1$$
$$c = \frac{\sqrt{3}}{2}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
$$w_{2} = - \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
cambiamos
$$2 \sin^{2}{\left(x \right)} - \sqrt{3} \sin{\left(x \right)} - \sin{\left(x \right)} + \frac{\sqrt{3}}{2} = 0$$
$$\left(\left(2 \sin{\left(x \right)} - 1\right) \sin{\left(x \right)} - \sqrt{3} \sin{\left(x \right)}\right) + \frac{\sqrt{3}}{2} = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Abramos la expresión en la ecuación
$$w \left(2 w - 1\right) - \sqrt{3} w + \frac{\sqrt{3}}{2} = 0$$
Obtenemos la ecuación cuadrática
$$2 w^{2} - \sqrt{3} w - w + \frac{\sqrt{3}}{2} = 0$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 2$$
$$b = - \sqrt{3} - 1$$
$$c = \frac{\sqrt{3}}{2}$$
, entonces
D = b^2 - 4 * a * c =
(-1 - sqrt(3))^2 - 4 * (2) * (sqrt(3)/2) = (-1 - sqrt(3))^2 - 4*sqrt(3)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
$$w_{2} = - \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4}$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{- 4 \sqrt{3} + \left(- \sqrt{3} - 1\right)^{2}}}{4} + \frac{1}{4} + \frac{\sqrt{3}}{4} \right)} + \pi$$
Respuesta rápida
[src]
pi
x1 = --
6
$$x_{1} = \frac{\pi}{6}$$
pi
x2 = --
3
$$x_{2} = \frac{\pi}{3}$$
2*pi
x3 = ----
3
$$x_{3} = \frac{2 \pi}{3}$$
5*pi
x4 = ----
6
$$x_{4} = \frac{5 \pi}{6}$$
x4 = 5*pi/6
Suma y producto de raíces
[src]
suma
pi pi 2*pi 5*pi -- + -- + ---- + ---- 6 3 3 6
$$\frac{5 \pi}{6} + \left(\left(\frac{\pi}{6} + \frac{\pi}{3}\right) + \frac{2 \pi}{3}\right)$$
=
2*pi
$$2 \pi$$
producto
pi pi 2*pi 5*pi --*--*----*---- 6 3 3 6
$$\frac{5 \pi}{6} \frac{2 \pi}{3} \frac{\pi}{6} \frac{\pi}{3}$$
=
4 5*pi ----- 162
$$\frac{5 \pi^{4}}{162}$$
5*pi^4/162
Respuesta numérica
[src]
x1 = -12.0427718387609
x2 = -68.0678408277789
x3 = 44.5058959258554
x4 = -79.5870138909414
x5 = 63.8790506229925
x6 = 82.7286065445312
x7 = -85.870199198121
x8 = -22.5147473507269
x9 = 46.6002910282486
x10 = 34.0339204138894
x11 = -35.6047167406843
x12 = 2.61799387799149
x13 = -5.75958653158129
x14 = 65.4498469497874
x15 = -41.3643032722656
x16 = 2.0943951023932
x17 = 52.8834763354282
x18 = 70.162235930172
x19 = 38.7463093942741
x20 = 76.4454212373516
x21 = 50.789081233035
x22 = -73.3038285837618
x23 = 90.5825881785057
x24 = -18.3259571459405
x25 = 71.733032256967
x26 = -60.2138591938044
x27 = -81.1578102177363
x28 = -72.7802298081635
x29 = 31.9395253114962
x30 = -24.0855436775217
x31 = 84.2994028713261
x32 = 78.0162175641465
x33 = -30.8923277602996
x34 = -93.7241808320955
x35 = -41.8879020478639
x36 = -3.66519142918809
x37 = 90.0589894029074
x38 = -66.497044500984
x39 = 26.1799387799149
x40 = -56.025068989018
x41 = -62.3082542961976
x42 = 14.6607657167524
x43 = -110.479341651241
x44 = -87.4409955249159
x45 = -61.7846555205993
x46 = -85.3466004225227
x47 = 27.7507351067098
x48 = -49.7418836818384
x49 = -24.60914245312
x50 = 77.4926187885482
x51 = 8.37758040957278
x52 = -100.007366139275
x53 = -9.94837673636768
x54 = 52.3598775598299
x55 = -97.9129710368819
x56 = -68.5914396033772
x57 = 38.2227106186758
x58 = 69.6386371545737
x59 = -35.081117965086
x60 = 0.523598775598299
x61 = 59.1666616426078
x62 = -43.4586983746588
x63 = 96.342174710087
x64 = 82.2050077689329
x65 = -54.4542726622231
x66 = -28.7979326579064
x67 = -79.0634151153431
x68 = 96.8657734856853
x69 = 19.8967534727354
x70 = 25.6563400043166
x71 = -173.834793498635
x72 = -53.9306738866248
x73 = 8.90117918517108
x74 = -10.471975511966
x75 = -74.8746249105567
x76 = 19.3731546971371
x77 = -389.033890269536
x78 = 63.3554518473942
x79 = 57.0722665402146
x80 = -16.2315620435473
x81 = 15.1843644923507
x82 = 40.317105721069
x83 = -37.1755130674792
x84 = 101.054563690472
x85 = -48.1710873550435
x86 = 46.0766922526503
x87 = -29.3215314335047
x88 = -47.6474885794452
x89 = 13.0899693899575
x90 = 6.80678408277789
x91 = -17.8023583703422
x92 = 94.7713783832921
x93 = -92.1533845053006
x94 = 88.4881930761125
x95 = 21.4675497995303
x95 = 21.4675497995303