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x^3+3*x^2+4=0

x^3+3*x^2+4=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
 3      2        
x  + 3*x  + 4 = 0
$$\left(x^{3} + 3 x^{2}\right) + 4 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 4$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 4$$
Gráfica
Respuesta rápida [src]
             _______________                            /                                  _______________\
          3 /           ___                             |            ___            ___ 3 /           ___ |
          \/  81 + 54*\/ 2              3               |        3*\/ 3           \/ 3 *\/  81 + 54*\/ 2  |
x1 = -1 + ------------------ + -------------------- + I*|- -------------------- + ------------------------|
                  6                 _______________     |       _______________              6            |
                                 3 /           ___      |    3 /           ___                            |
                               2*\/  81 + 54*\/ 2       \  2*\/  81 + 54*\/ 2                             /
$$x_{1} = -1 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)$$
             _______________                            /           _______________                       \
          3 /           ___                             |    ___ 3 /           ___              ___       |
          \/  81 + 54*\/ 2              3               |  \/ 3 *\/  81 + 54*\/ 2           3*\/ 3        |
x2 = -1 + ------------------ + -------------------- + I*|- ------------------------ + --------------------|
                  6                 _______________     |             6                    _______________|
                                 3 /           ___      |                               3 /           ___ |
                               2*\/  81 + 54*\/ 2       \                             2*\/  81 + 54*\/ 2  /
$$x_{2} = -1 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right)$$
                                  _______________
                               3 /           ___ 
                  3            \/  81 + 54*\/ 2  
x3 = -1 - ------------------ - ------------------
             _______________           3         
          3 /           ___                      
          \/  81 + 54*\/ 2                       
$$x_{3} = - \frac{\sqrt[3]{54 \sqrt{2} + 81}}{3} - 1 - \frac{3}{\sqrt[3]{54 \sqrt{2} + 81}}$$
x3 = -(54*sqrt(2) + 81)^(1/3)/3 - 1 - 3/(54*sqrt(2) + 81)^(1/3)
Suma y producto de raíces [src]
suma
        _______________                            /                                  _______________\           _______________                            /           _______________                       \                                _______________
     3 /           ___                             |            ___            ___ 3 /           ___ |        3 /           ___                             |    ___ 3 /           ___              ___       |                             3 /           ___ 
     \/  81 + 54*\/ 2              3               |        3*\/ 3           \/ 3 *\/  81 + 54*\/ 2  |        \/  81 + 54*\/ 2              3               |  \/ 3 *\/  81 + 54*\/ 2           3*\/ 3        |                3            \/  81 + 54*\/ 2  
-1 + ------------------ + -------------------- + I*|- -------------------- + ------------------------| + -1 + ------------------ + -------------------- + I*|- ------------------------ + --------------------| + -1 - ------------------ - ------------------
             6                 _______________     |       _______________              6            |                6                 _______________     |             6                    _______________|           _______________           3         
                            3 /           ___      |    3 /           ___                            |                               3 /           ___      |                               3 /           ___ |        3 /           ___                      
                          2*\/  81 + 54*\/ 2       \  2*\/  81 + 54*\/ 2                             /                             2*\/  81 + 54*\/ 2       \                             2*\/  81 + 54*\/ 2  /        \/  81 + 54*\/ 2                       
$$\left(- \frac{\sqrt[3]{54 \sqrt{2} + 81}}{3} - 1 - \frac{3}{\sqrt[3]{54 \sqrt{2} + 81}}\right) + \left(\left(-1 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right)\right) + \left(-1 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)\right)\right)$$
=
       /                                  _______________\     /           _______________                       \
       |            ___            ___ 3 /           ___ |     |    ___ 3 /           ___              ___       |
       |        3*\/ 3           \/ 3 *\/  81 + 54*\/ 2  |     |  \/ 3 *\/  81 + 54*\/ 2           3*\/ 3        |
-3 + I*|- -------------------- + ------------------------| + I*|- ------------------------ + --------------------|
       |       _______________              6            |     |             6                    _______________|
       |    3 /           ___                            |     |                               3 /           ___ |
       \  2*\/  81 + 54*\/ 2                             /     \                             2*\/  81 + 54*\/ 2  /
$$-3 + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right) + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)$$
producto
/        _______________                            /                                  _______________\\ /        _______________                            /           _______________                       \\ /                             _______________\
|     3 /           ___                             |            ___            ___ 3 /           ___ || |     3 /           ___                             |    ___ 3 /           ___              ___       || |                          3 /           ___ |
|     \/  81 + 54*\/ 2              3               |        3*\/ 3           \/ 3 *\/  81 + 54*\/ 2  || |     \/  81 + 54*\/ 2              3               |  \/ 3 *\/  81 + 54*\/ 2           3*\/ 3        || |             3            \/  81 + 54*\/ 2  |
|-1 + ------------------ + -------------------- + I*|- -------------------- + ------------------------||*|-1 + ------------------ + -------------------- + I*|- ------------------------ + --------------------||*|-1 - ------------------ - ------------------|
|             6                 _______________     |       _______________              6            || |             6                 _______________     |             6                    _______________|| |        _______________           3         |
|                            3 /           ___      |    3 /           ___                            || |                            3 /           ___      |                               3 /           ___ || |     3 /           ___                      |
\                          2*\/  81 + 54*\/ 2       \  2*\/  81 + 54*\/ 2                             // \                          2*\/  81 + 54*\/ 2       \                             2*\/  81 + 54*\/ 2  // \     \/  81 + 54*\/ 2                       /
$$\left(-1 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)\right) \left(-1 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right)\right) \left(- \frac{\sqrt[3]{54 \sqrt{2} + 81}}{3} - 1 - \frac{3}{\sqrt[3]{54 \sqrt{2} + 81}}\right)$$
=
-4
$$-4$$
-4
Respuesta numérica [src]
x1 = 0.17765069880406 - 1.07730381284996*i
x2 = 0.17765069880406 + 1.07730381284996*i
x3 = -3.35530139760812
x3 = -3.35530139760812
Gráfico
x^3+3*x^2+4=0 la ecuación