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-x^3+x^2-16*x-20=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   3    2                
- x  + x  - 16*x - 20 = 0
(−16x+(−x3+x2))−20=0\left(- 16 x + \left(- x^{3} + x^{2}\right)\right) - 20 = 0
Simplificar
Teorema de Cardano-Vieta
reescribamos la ecuación
(−16x+(−x3+x2))−20=0\left(- 16 x + \left(- x^{3} + x^{2}\right)\right) - 20 = 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−x2+16x+20=0x^{3} - x^{2} + 16 x + 20 = 0
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=−1p = -1
q=caq = \frac{c}{a}
q=16q = 16
v=dav = \frac{d}{a}
v=20v = 20
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=1x_{1} + x_{2} + x_{3} = 1
x1x2+x1x3+x2x3=16x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 16
x1x2x3=20x_{1} x_{2} x_{3} = 20
Gráfica
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-25002500
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
                                 __________________     /         __________________                          \                                    __________________     /                                     __________________\          __________________                          
                              3 /           ______      |  ___ 3 /           ______                 ___       |                                 3 /           ______      |               ___            ___ 3 /           ______ |       3 /           ______                           
1              47             \/  341 + 6*\/ 6114       |\/ 3 *\/  341 + 6*\/ 6114             47*\/ 3        |   1              47             \/  341 + 6*\/ 6114       |          47*\/ 3           \/ 3 *\/  341 + 6*\/ 6114  |   1   \/  341 + 6*\/ 6114                47          
- - ----------------------- + --------------------- + I*|--------------------------- + -----------------------| + - - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------| + - - --------------------- + -----------------------
3        __________________             6               |             6                     __________________|   3        __________________             6               |       __________________                6             |   3             3                  __________________
      3 /           ______                              |                                3 /           ______ |         3 /           ______                              |    3 /           ______                               |                                 3 /           ______ 
    6*\/  341 + 6*\/ 6114                               \                              6*\/  341 + 6*\/ 6114  /       6*\/  341 + 6*\/ 6114                               \  6*\/  341 + 6*\/ 6114                                /                               3*\/  341 + 6*\/ 6114
(−341+6611433+13+473341+661143)+((−476341+661143+13+341+6611436+i(−3341+6611436−4736341+661143))+(−476341+661143+13+341+6611436+i(4736341+661143+3341+6611436)))\left(- \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{3} + \frac{1}{3} + \frac{47}{3 \sqrt[3]{341 + 6 \sqrt{6114}}}\right) + \left(\left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right)\right) + \left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right)\right)\right) 
=
      /                                     __________________\     /         __________________                          \
      |               ___            ___ 3 /           ______ |     |  ___ 3 /           ______                 ___       |
      |          47*\/ 3           \/ 3 *\/  341 + 6*\/ 6114  |     |\/ 3 *\/  341 + 6*\/ 6114             47*\/ 3        |
1 + I*|- ----------------------- - ---------------------------| + I*|--------------------------- + -----------------------|
      |       __________________                6             |     |             6                     __________________|
      |    3 /           ______                               |     |                                3 /           ______ |
      \  6*\/  341 + 6*\/ 6114                                /     \                              6*\/  341 + 6*\/ 6114  /
1+i(−3341+6611436−4736341+661143)+i(4736341+661143+3341+6611436)1 + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right) + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right) 
producto
/                                 __________________     /         __________________                          \\ /                                 __________________     /                                     __________________\\ /       __________________                          \
|                              3 /           ______      |  ___ 3 /           ______                 ___       || |                              3 /           ______      |               ___            ___ 3 /           ______ || |    3 /           ______                           |
|1              47             \/  341 + 6*\/ 6114       |\/ 3 *\/  341 + 6*\/ 6114             47*\/ 3        || |1              47             \/  341 + 6*\/ 6114       |          47*\/ 3           \/ 3 *\/  341 + 6*\/ 6114  || |1   \/  341 + 6*\/ 6114                47          |
|- - ----------------------- + --------------------- + I*|--------------------------- + -----------------------||*|- - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------||*|- - --------------------- + -----------------------|
|3        __________________             6               |             6                     __________________|| |3        __________________             6               |       __________________                6             || |3             3                  __________________|
|      3 /           ______                              |                                3 /           ______ || |      3 /           ______                              |    3 /           ______                               || |                              3 /           ______ |
\    6*\/  341 + 6*\/ 6114                               \                              6*\/  341 + 6*\/ 6114  // \    6*\/  341 + 6*\/ 6114                               \  6*\/  341 + 6*\/ 6114                                // \                            3*\/  341 + 6*\/ 6114  /
(−476341+661143+13+341+6611436+i(4736341+661143+3341+6611436))(−476341+661143+13+341+6611436+i(−3341+6611436−4736341+661143))(−341+6611433+13+473341+661143)\left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right)\right) \left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right)\right) \left(- \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{3} + \frac{1}{3} + \frac{47}{3 \sqrt[3]{341 + 6 \sqrt{6114}}}\right) 
=
-20
−20-20 
-20
Respuesta rápida [src] 
                                      __________________     /         __________________                          \
                                   3 /           ______      |  ___ 3 /           ______                 ___       |
     1              47             \/  341 + 6*\/ 6114       |\/ 3 *\/  341 + 6*\/ 6114             47*\/ 3        |
x1 = - - ----------------------- + --------------------- + I*|--------------------------- + -----------------------|
     3        __________________             6               |             6                     __________________|
           3 /           ______                              |                                3 /           ______ |
         6*\/  341 + 6*\/ 6114                               \                              6*\/  341 + 6*\/ 6114  /
x1=−476341+661143+13+341+6611436+i(4736341+661143+3341+6611436)x_{1} = - \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right) 
                                      __________________     /                                     __________________\
                                   3 /           ______      |               ___            ___ 3 /           ______ |
     1              47             \/  341 + 6*\/ 6114       |          47*\/ 3           \/ 3 *\/  341 + 6*\/ 6114  |
x2 = - - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------|
     3        __________________             6               |       __________________                6             |
           3 /           ______                              |    3 /           ______                               |
         6*\/  341 + 6*\/ 6114                               \  6*\/  341 + 6*\/ 6114                                /
x2=−476341+661143+13+341+6611436+i(−3341+6611436−4736341+661143)x_{2} = - \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right) 
            __________________                          
         3 /           ______                           
     1   \/  341 + 6*\/ 6114                47          
x3 = - - --------------------- + -----------------------
     3             3                  __________________
                                   3 /           ______ 
                                 3*\/  341 + 6*\/ 6114
x3=−341+6611433+13+473341+661143x_{3} = - \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{3} + \frac{1}{3} + \frac{47}{3 \sqrt[3]{341 + 6 \sqrt{6114}}} 
x3 = -(341 + 6*sqrt(6114))^(1/3)/3 + 1/3 + 47/(3*(341 + 6*sqrt(6114))^(1/3))
Respuesta numérica [src] 
x1 = 1.0467663862312 - 4.1465198455534*i
x2 = 1.0467663862312 + 4.1465198455534*i
x3 = -1.0935327724624
x3 = -1.0935327724624
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