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18*x^3-16*x^2+550*x+144=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
    3       2                  
18*x  - 16*x  + 550*x + 144 = 0
(550x+(18x3−16x2))+144=0\left(550 x + \left(18 x^{3} - 16 x^{2}\right)\right) + 144 = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(550x+(18x3−16x2))+144=0\left(550 x + \left(18 x^{3} - 16 x^{2}\right)\right) + 144 = 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
x3−8x29+275x9+8=0x^{3} - \frac{8 x^{2}}{9} + \frac{275 x}{9} + 8 = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=−89p = - \frac{8}{9}
q=caq = \frac{c}{a}
q=2759q = \frac{275}{9}
v=dav = \frac{d}{a}
v=8v = 8
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=89x_{1} + x_{2} + x_{3} = \frac{8}{9}
x1x2+x1x3+x2x3=2759x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{275}{9}
x1x2x3=8x_{1} x_{2} x_{3} = 8
Suma y producto de raíces [src] 
suma
                                              ________________________     /           ________________________                                    \                                                 ________________________     /                                                 ________________________\             ________________________                                    
                                             /            ___________      |          /            ___________                                     |                                                /            ___________      |                                                /            ___________ |            /            ___________                                     
                                            /  167320   \/ 585523689       |  ___    /  167320   \/ 585523689                                      |                                               /  167320   \/ 585523689       |                                        ___    /  167320   \/ 585523689  |           /  167320   \/ 585523689                                      
                                         3 /   ------ + -------------      |\/ 3 *3 /   ------ + -------------                       ___           |                                            3 /   ------ + -------------      |                     ___              \/ 3 *3 /   ------ + ------------- |        3 /   ------ + -------------                                     
8                   7361                 \/     729           27           |      \/     729           27                     7361*\/ 3            |   8                   7361                 \/     729           27           |              7361*\/ 3                     \/     729           27      |   8    \/     729           27                        7361              
-- - --------------------------------- + ----------------------------- + I*|----------------------------------- + ---------------------------------| + -- - --------------------------------- + ----------------------------- + I*|- --------------------------------- - -----------------------------------| + -- - ----------------------------- + ---------------------------------
27            ________________________                 6                   |                 6                             ________________________|   27            ________________________                 6                   |           ________________________                    6                 |   27                 3                          ________________________
             /            ___________                                      |                                              /            ___________ |                /            ___________                                      |          /            ___________                                       |                                                /            ___________ 
            /  167320   \/ 585523689                                       |                                             /  167320   \/ 585523689  |               /  167320   \/ 585523689                                       |         /  167320   \/ 585523689                                        |                                               /  167320   \/ 585523689  
     486*3 /   ------ + -------------                                      |                                      486*3 /   ------ + ------------- |        486*3 /   ------ + -------------                                      |  486*3 /   ------ + -------------                                       |                                        243*3 /   ------ + ------------- 
         \/     729           27                                           \                                          \/     729           27      /            \/     729           27                                           \      \/     729           27                                            /                                            \/     729           27
(−167320729+5855236892733+827+7361243167320729+585523689273)+((−7361486167320729+585523689273+827+167320729+5855236892736+i(−3167320729+5855236892736−73613486167320729+585523689273))+(−7361486167320729+585523689273+827+167320729+5855236892736+i(73613486167320729+585523689273+3167320729+5855236892736)))\left(- \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{3} + \frac{8}{27} + \frac{7361}{243 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right) + \left(\left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right)\right) + \left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right)\right)\right) 
=
      /                                                 ________________________\     /           ________________________                                    \
      |                                                /            ___________ |     |          /            ___________                                     |
      |                                        ___    /  167320   \/ 585523689  |     |  ___    /  167320   \/ 585523689                                      |
      |                     ___              \/ 3 *3 /   ------ + ------------- |     |\/ 3 *3 /   ------ + -------------                       ___           |
8     |              7361*\/ 3                     \/     729           27      |     |      \/     729           27                     7361*\/ 3            |
- + I*|- --------------------------------- - -----------------------------------| + I*|----------------------------------- + ---------------------------------|
9     |           ________________________                    6                 |     |                 6                             ________________________|
      |          /            ___________                                       |     |                                              /            ___________ |
      |         /  167320   \/ 585523689                                        |     |                                             /  167320   \/ 585523689  |
      |  486*3 /   ------ + -------------                                       |     |                                      486*3 /   ------ + ------------- |
      \      \/     729           27                                            /     \                                          \/     729           27      /
89+i(−3167320729+5855236892736−73613486167320729+585523689273)+i(73613486167320729+585523689273+3167320729+5855236892736)\frac{8}{9} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right) + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right) 
producto
/                                              ________________________     /           ________________________                                    \\ /                                              ________________________     /                                                 ________________________\\ /          ________________________                                    \
|                                             /            ___________      |          /            ___________                                     || |                                             /            ___________      |                                                /            ___________ || |         /            ___________                                     |
|                                            /  167320   \/ 585523689       |  ___    /  167320   \/ 585523689                                      || |                                            /  167320   \/ 585523689       |                                        ___    /  167320   \/ 585523689  || |        /  167320   \/ 585523689                                      |
|                                         3 /   ------ + -------------      |\/ 3 *3 /   ------ + -------------                       ___           || |                                         3 /   ------ + -------------      |                     ___              \/ 3 *3 /   ------ + ------------- || |     3 /   ------ + -------------                                     |
|8                   7361                 \/     729           27           |      \/     729           27                     7361*\/ 3            || |8                   7361                 \/     729           27           |              7361*\/ 3                     \/     729           27      || |8    \/     729           27                        7361              |
|-- - --------------------------------- + ----------------------------- + I*|----------------------------------- + ---------------------------------||*|-- - --------------------------------- + ----------------------------- + I*|- --------------------------------- - -----------------------------------||*|-- - ----------------------------- + ---------------------------------|
|27            ________________________                 6                   |                 6                             ________________________|| |27            ________________________                 6                   |           ________________________                    6                 || |27                 3                          ________________________|
|             /            ___________                                      |                                              /            ___________ || |             /            ___________                                      |          /            ___________                                       || |                                             /            ___________ |
|            /  167320   \/ 585523689                                       |                                             /  167320   \/ 585523689  || |            /  167320   \/ 585523689                                       |         /  167320   \/ 585523689                                        || |                                            /  167320   \/ 585523689  |
|     486*3 /   ------ + -------------                                      |                                      486*3 /   ------ + ------------- || |     486*3 /   ------ + -------------                                      |  486*3 /   ------ + -------------                                       || |                                     243*3 /   ------ + ------------- |
\         \/     729           27                                           \                                          \/     729           27      // \         \/     729           27                                           \      \/     729           27                                            // \                                         \/     729           27      /
(−7361486167320729+585523689273+827+167320729+5855236892736+i(73613486167320729+585523689273+3167320729+5855236892736))(−7361486167320729+585523689273+827+167320729+5855236892736+i(−3167320729+5855236892736−73613486167320729+585523689273))(−167320729+5855236892733+827+7361243167320729+585523689273)\left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right)\right) \left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{3} + \frac{8}{27} + \frac{7361}{243 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right) 
=
-8
−8-8 
-8
Respuesta rápida [src] 
                                                   ________________________     /           ________________________                                    \
                                                  /            ___________      |          /            ___________                                     |
                                                 /  167320   \/ 585523689       |  ___    /  167320   \/ 585523689                                      |
                                              3 /   ------ + -------------      |\/ 3 *3 /   ------ + -------------                       ___           |
     8                   7361                 \/     729           27           |      \/     729           27                     7361*\/ 3            |
x1 = -- - --------------------------------- + ----------------------------- + I*|----------------------------------- + ---------------------------------|
     27            ________________________                 6                   |                 6                             ________________________|
                  /            ___________                                      |                                              /            ___________ |
                 /  167320   \/ 585523689                                       |                                             /  167320   \/ 585523689  |
          486*3 /   ------ + -------------                                      |                                      486*3 /   ------ + ------------- |
              \/     729           27                                           \                                          \/     729           27      /
x1=−7361486167320729+585523689273+827+167320729+5855236892736+i(73613486167320729+585523689273+3167320729+5855236892736)x_{1} = - \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right) 
                                                   ________________________     /                                                 ________________________\
                                                  /            ___________      |                                                /            ___________ |
                                                 /  167320   \/ 585523689       |                                        ___    /  167320   \/ 585523689  |
                                              3 /   ------ + -------------      |                     ___              \/ 3 *3 /   ------ + ------------- |
     8                   7361                 \/     729           27           |              7361*\/ 3                     \/     729           27      |
x2 = -- - --------------------------------- + ----------------------------- + I*|- --------------------------------- - -----------------------------------|
     27            ________________________                 6                   |           ________________________                    6                 |
                  /            ___________                                      |          /            ___________                                       |
                 /  167320   \/ 585523689                                       |         /  167320   \/ 585523689                                        |
          486*3 /   ------ + -------------                                      |  486*3 /   ------ + -------------                                       |
              \/     729           27                                           \      \/     729           27                                            /
x2=−7361486167320729+585523689273+827+167320729+5855236892736+i(−3167320729+5855236892736−73613486167320729+585523689273)x_{2} = - \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right) 
               ________________________                                    
              /            ___________                                     
             /  167320   \/ 585523689                                      
          3 /   ------ + -------------                                     
     8    \/     729           27                        7361              
x3 = -- - ----------------------------- + ---------------------------------
     27                 3                          ________________________
                                                  /            ___________ 
                                                 /  167320   \/ 585523689  
                                          243*3 /   ------ + ------------- 
                                              \/     729           27
x3=−167320729+5855236892733+827+7361243167320729+585523689273x_{3} = - \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{3} + \frac{8}{27} + \frac{7361}{243 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} 
x3 = -(167320/729 + sqrt(585523689)/27)^(1/3)/3 + 8/27 + 7361/(243*(167320/729 + sqrt(585523689)/27)^(1/3))
Respuesta numérica [src] 
x1 = -0.259291807789602
x2 = 0.574090348339245 - 5.52482485478139*i
x3 = 0.574090348339245 + 5.52482485478139*i
x3 = 0.574090348339245 + 5.52482485478139*i
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