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(-v^3+2*sin(x))*(-2*x^2-3*x+2)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
/   3           \ /     2          \    
\- v  + 2*sin(x)/*\- 2*x  - 3*x + 2/ = 0
(−v3+2sin⁡(x))((−2x2−3x)+2)=0\left(- v^{3} + 2 \sin{\left(x \right)}\right) \left(\left(- 2 x^{2} - 3 x\right) + 2\right) = 0
Simplificar
Solución detallada
Tenemos la ecuación
(−v3+2sin⁡(x))((−2x2−3x)+2)=0\left(- v^{3} + 2 \sin{\left(x \right)}\right) \left(\left(- 2 x^{2} - 3 x\right) + 2\right) = 0
cambiamos
(v3−2sin⁡(x))(2x2+3x−2)=0\left(v^{3} - 2 \sin{\left(x \right)}\right) \left(2 x^{2} + 3 x - 2\right) = 0
(−v3+2sin⁡(x))((−2x2−3x)+2)=0\left(- v^{3} + 2 \sin{\left(x \right)}\right) \left(\left(- 2 x^{2} - 3 x\right) + 2\right) = 0
Sustituimos
w=sin⁡(x)w = \sin{\left(x \right)}
Tenemos la ecuación:
(-v^3 + 2*w)*(-2*x^2 - 3*x + 2) = 0

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

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

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

Obtenemos la respuesta: w = v^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⁡(v32)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{v^{3}}{2} \right)}
x1=2πn+asin⁡(v32)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{v^{3}}{2} \right)}
x2=2πn−asin⁡(w1)+πx_{2} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x2=2πn−asin⁡(v32)+πx_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{v^{3}}{2} \right)} + \pi
x2=2πn−asin⁡(v32)+πx_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{v^{3}}{2} \right)} + \pi
Gráfica
Respuesta rápida [src] 
x1 = -2
x1=−2x_{1} = -2 
x2 = 1/2
x2=12x_{2} = \frac{1}{2} 
            /|            _________|\                             
            ||   3       /       6 ||      /  /        _________\\
            ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 ||
x3 = - I*log||---- - --------------|| + arg\I*\v  - \/  -4 + v  //
            \| 2           2       |/
x3=−ilog⁡(∣iv32−iv6−42∣)+arg⁡(i(v3−v6−4))x_{3} = - i \log{\left(\left|{\frac{i v^{3}}{2} - \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} - \sqrt{v^{6} - 4}\right) \right)} 
            /|            _________|\                             
            ||   3       /       6 ||      /  /        _________\\
            ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 ||
x4 = - I*log||---- + --------------|| + arg\I*\v  + \/  -4 + v  //
            \| 2           2       |/
x4=−ilog⁡(∣iv32+iv6−42∣)+arg⁡(i(v3+v6−4))x_{4} = - i \log{\left(\left|{\frac{i v^{3}}{2} + \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} + \sqrt{v^{6} - 4}\right) \right)} 
x4 = -i*log(Abs(i*v^3/2 + i*sqrt(v^6 - 4)/2)) + arg(i*(v^3 + sqrt(v^6 - 4)))
Suma y producto de raíces [src] 
suma
                  /|            _________|\                                       /|            _________|\                             
                  ||   3       /       6 ||      /  /        _________\\          ||   3       /       6 ||      /  /        _________\\
                  ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 ||          ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 ||
-2 + 1/2 + - I*log||---- - --------------|| + arg\I*\v  - \/  -4 + v  // + - I*log||---- + --------------|| + arg\I*\v  + \/  -4 + v  //
                  \| 2           2       |/                                       \| 2           2       |/
(−ilog⁡(∣iv32+iv6−42∣)+arg⁡(i(v3+v6−4)))+((−ilog⁡(∣iv32−iv6−42∣)+arg⁡(i(v3−v6−4)))+(−2+12))\left(- i \log{\left(\left|{\frac{i v^{3}}{2} + \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} + \sqrt{v^{6} - 4}\right) \right)}\right) + \left(\left(- i \log{\left(\left|{\frac{i v^{3}}{2} - \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} - \sqrt{v^{6} - 4}\right) \right)}\right) + \left(-2 + \frac{1}{2}\right)\right) 
=
           /|            _________|\        /|            _________|\                                                          
           ||   3       /       6 ||        ||   3       /       6 ||      /  /        _________\\      /  /        _________\\
  3        ||I*v    I*\/  -4 + v  ||        ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 ||      |  | 3     /       6 ||
- - - I*log||---- + --------------|| - I*log||---- - --------------|| + arg\I*\v  + \/  -4 + v  // + arg\I*\v  - \/  -4 + v  //
  2        \| 2           2       |/        \| 2           2       |/
−ilog⁡(∣iv32−iv6−42∣)−ilog⁡(∣iv32+iv6−42∣)+arg⁡(i(v3−v6−4))+arg⁡(i(v3+v6−4))−32- i \log{\left(\left|{\frac{i v^{3}}{2} - \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} - i \log{\left(\left|{\frac{i v^{3}}{2} + \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} - \sqrt{v^{6} - 4}\right) \right)} + \arg{\left(i \left(v^{3} + \sqrt{v^{6} - 4}\right) \right)} - \frac{3}{2} 
producto
    /       /|            _________|\                             \ /       /|            _________|\                             \
    |       ||   3       /       6 ||      /  /        _________\\| |       ||   3       /       6 ||      /  /        _________\\|
-2  |       ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 ||| |       ||I*v    I*\/  -4 + v  ||      |  | 3     /       6 |||
---*|- I*log||---- - --------------|| + arg\I*\v  - \/  -4 + v  //|*|- I*log||---- + --------------|| + arg\I*\v  + \/  -4 + v  //|
 2  \       \| 2           2       |/                             / \       \| 2           2       |/                             /
−1(−ilog⁡(∣iv32−iv6−42∣)+arg⁡(i(v3−v6−4)))(−ilog⁡(∣iv32+iv6−42∣)+arg⁡(i(v3+v6−4)))- 1 \left(- i \log{\left(\left|{\frac{i v^{3}}{2} - \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} - \sqrt{v^{6} - 4}\right) \right)}\right) \left(- i \log{\left(\left|{\frac{i v^{3}}{2} + \frac{i \sqrt{v^{6} - 4}}{2}}\right| \right)} + \arg{\left(i \left(v^{3} + \sqrt{v^{6} - 4}\right) \right)}\right) 
=
 /                                    /|        _________|\\ /                                    /|        _________|\\
 |     /  /        _________\\        || 3     /       6 ||| |     /  /        _________\\        || 3     /       6 |||
 |     |  | 3     /       6 ||        ||v  + \/  -4 + v  ||| |     |  | 3     /       6 ||        ||v  - \/  -4 + v  |||
-|- arg\I*\v  + \/  -4 + v  // + I*log|-------------------||*|- arg\I*\v  - \/  -4 + v  // + I*log|-------------------||
 \                                    \         2         // \                                    \         2         //
−(ilog⁡(∣v3−v6−4∣2)−arg⁡(i(v3−v6−4)))(ilog⁡(∣v3+v6−4∣2)−arg⁡(i(v3+v6−4)))- \left(i \log{\left(\frac{\left|{v^{3} - \sqrt{v^{6} - 4}}\right|}{2} \right)} - \arg{\left(i \left(v^{3} - \sqrt{v^{6} - 4}\right) \right)}\right) \left(i \log{\left(\frac{\left|{v^{3} + \sqrt{v^{6} - 4}}\right|}{2} \right)} - \arg{\left(i \left(v^{3} + \sqrt{v^{6} - 4}\right) \right)}\right) 
-(-arg(i*(v^3 + sqrt(-4 + v^6))) + i*log(Abs(v^3 + sqrt(-4 + v^6))/2))*(-arg(i*(v^3 - sqrt(-4 + v^6))) + i*log(Abs(v^3 - sqrt(-4 + v^6))/2))
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