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6x^3+3sqrt(2)*x^2-6*x+2sqrt(2)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   3       ___  2             ___    
6*x  + 3*\/ 2 *x  - 6*x + 2*\/ 2  = 0
(−6x+(6x3+32x2))+22=0\left(- 6 x + \left(6 x^{3} + 3 \sqrt{2} x^{2}\right)\right) + 2 \sqrt{2} = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(−6x+(6x3+32x2))+22=0\left(- 6 x + \left(6 x^{3} + 3 \sqrt{2} x^{2}\right)\right) + 2 \sqrt{2} = 0
de
ax3+bx2+cx+d=0a x^{3} + b x^{2} + c x + d = 0
como ecuación cúbica reducida
x3+bx2a+cxa+da=0x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0
x3+2x22−x+23=0x^{3} + \frac{\sqrt{2} x^{2}}{2} - x + \frac{\sqrt{2}}{3} = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=22p = \frac{\sqrt{2}}{2}
q=caq = \frac{c}{a}
q=−1q = -1
v=dav = \frac{d}{a}
v=23v = \frac{\sqrt{2}}{3}
Fórmulas de Cardano-Vieta
x1+x2+x3=−px_{1} + x_{2} + x_{3} = - p
x1x2+x1x3+x2x3=qx_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q
x1x2x3=vx_{1} x_{2} x_{3} = v
x1+x2+x3=−22x_{1} + x_{2} + x_{3} = - \frac{\sqrt{2}}{2}
x1x2+x1x3+x2x3=−1x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1
x1x2x3=23x_{1} x_{2} x_{3} = \frac{\sqrt{2}}{3}
Gráfica
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-1000010000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
         ___     /    ___ 3 ___     ___  2/3\     ___ 3 ___     ___  2/3
       \/ 2      |  \/ 6 *\/ 7    \/ 6 *7   |   \/ 2 *\/ 7    \/ 2 *7   
x1 = - ----- + I*|- ----------- + ----------| + ----------- + ----------
         6       \       12           12    /        12           12
x1=−26+27312+2⋅72312+i(−67312+6⋅72312)x_{1} = - \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right) 
         ___     /    ___  2/3     ___ 3 ___\     ___ 3 ___     ___  2/3
       \/ 2      |  \/ 6 *7      \/ 6 *\/ 7 |   \/ 2 *\/ 7    \/ 2 *7   
x2 = - ----- + I*|- ---------- + -----------| + ----------- + ----------
         6       \      12            12    /        12           12
x2=−26+27312+2⋅72312+i(−6⋅72312+67312)x_{2} = - \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right) 
         ___     ___ 3 ___     ___    2/3
       \/ 2    \/ 2 *\/ 7    \/ 2 *4*7   
x3 = - ----- - ----------- - ------------
         6          6             24
x3=−2⋅4⋅72324−2736−26x_{3} = - \frac{\sqrt{2} \cdot 4 \cdot 7^{\frac{2}{3}}}{24} - \frac{\sqrt{2} \sqrt[3]{7}}{6} - \frac{\sqrt{2}}{6} 
x3 = -sqrt(2)*4*7^(2/3)/24 - sqrt(2)*7^(1/3)/6 - sqrt(2)/6
Suma y producto de raíces [src] 
suma
    ___     /    ___ 3 ___     ___  2/3\     ___ 3 ___     ___  2/3       ___     /    ___  2/3     ___ 3 ___\     ___ 3 ___     ___  2/3       ___     ___ 3 ___     ___    2/3
  \/ 2      |  \/ 6 *\/ 7    \/ 6 *7   |   \/ 2 *\/ 7    \/ 2 *7        \/ 2      |  \/ 6 *7      \/ 6 *\/ 7 |   \/ 2 *\/ 7    \/ 2 *7        \/ 2    \/ 2 *\/ 7    \/ 2 *4*7   
- ----- + I*|- ----------- + ----------| + ----------- + ---------- + - ----- + I*|- ---------- + -----------| + ----------- + ---------- + - ----- - ----------- - ------------
    6       \       12           12    /        12           12           6       \      12            12    /        12           12           6          6             24
(−2⋅4⋅72324−2736−26)+((−26+27312+2⋅72312+i(−6⋅72312+67312))+(−26+27312+2⋅72312+i(−67312+6⋅72312)))\left(- \frac{\sqrt{2} \cdot 4 \cdot 7^{\frac{2}{3}}}{24} - \frac{\sqrt{2} \sqrt[3]{7}}{6} - \frac{\sqrt{2}}{6}\right) + \left(\left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right)\right) + \left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right)\right)\right) 
=
    ___     /    ___ 3 ___     ___  2/3\     /    ___  2/3     ___ 3 ___\
  \/ 2      |  \/ 6 *\/ 7    \/ 6 *7   |     |  \/ 6 *7      \/ 6 *\/ 7 |
- ----- + I*|- ----------- + ----------| + I*|- ---------- + -----------|
    2       \       12           12    /     \      12            12    /
−22+i(−6⋅72312+67312)+i(−67312+6⋅72312)- \frac{\sqrt{2}}{2} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right) + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right) 
producto
/    ___     /    ___ 3 ___     ___  2/3\     ___ 3 ___     ___  2/3\ /    ___     /    ___  2/3     ___ 3 ___\     ___ 3 ___     ___  2/3\ /    ___     ___ 3 ___     ___    2/3\
|  \/ 2      |  \/ 6 *\/ 7    \/ 6 *7   |   \/ 2 *\/ 7    \/ 2 *7   | |  \/ 2      |  \/ 6 *7      \/ 6 *\/ 7 |   \/ 2 *\/ 7    \/ 2 *7   | |  \/ 2    \/ 2 *\/ 7    \/ 2 *4*7   |
|- ----- + I*|- ----------- + ----------| + ----------- + ----------|*|- ----- + I*|- ---------- + -----------| + ----------- + ----------|*|- ----- - ----------- - ------------|
\    6       \       12           12    /        12           12    / \    6       \      12            12    /        12           12    / \    6          6             24     /
(−26+27312+2⋅72312+i(−67312+6⋅72312))(−26+27312+2⋅72312+i(−6⋅72312+67312))(−2⋅4⋅72324−2736−26)\left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right)\right) \left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right)\right) \left(- \frac{\sqrt{2} \cdot 4 \cdot 7^{\frac{2}{3}}}{24} - \frac{\sqrt{2} \sqrt[3]{7}}{6} - \frac{\sqrt{2}}{6}\right) 
=
   ___ 
-\/ 2  
-------
   3
−23- \frac{\sqrt{2}}{3} 
-sqrt(2)/3
Respuesta numérica [src] 
x1 = 0.42099215515006 + 0.356477207629844*i
x2 = 0.42099215515006 - 0.356477207629844*i
x3 = -1.54909109148667
x3 = -1.54909109148667
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