Sr Examen
Otras calculadoras
-
¿Cómo usar?
- La ecuación:
- Ecuación x(3x+1)-4=0
- Ecuación (u^2-12*u)/3=0
- Ecuación 49p+14pq+q^2=196
- Ecuación x^3+2016*x^2=1008
- Expresar {x} en función de y en la ecuación:
- -6*x-6*y=-12
- -19*x-2*y=2
- 12*x+1*y=11
- -14*x+6*y=-8
- Derivada de:
-
-4x^3
- Integral de d{x}:
- -4x^3
-
Expresiones idénticas
- -4x^ tres =5x+ sesenta y cuatro
- menos 4x al cubo es igual a 5x más 64
- menos 4x en el grado tres es igual a 5x más sesenta y cuatro
- -4x3=5x+64
- -4x³=5x+64
- -4x en el grado 3=5x+64
-
Expresiones semejantes
- -4x^3=5x-64
- 4x^3=5x+64
-4x^3=5x+64 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- 4 x^{3} = 5 x + 64$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{5 x}{4} + 16 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{4}$$
$$v = \frac{d}{a}$$
$$v = 16$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{5}{4}$$
$$x_{1} x_{2} x_{3} = 16$$
$$- 4 x^{3} = 5 x + 64$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{5 x}{4} + 16 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{4}$$
$$v = \frac{d}{a}$$
$$v = 16$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{5}{4}$$
$$x_{1} x_{2} x_{3} = 16$$
Suma y producto de raíces
[src]
suma
____________________ / ____________________ \ ____________________ / ____________________\ ____________________
/ ________ | / ________ | / ________ | / ________ | / ________
/ 3*\/ 332151 | ___ / 3*\/ 332151 | / 3*\/ 332151 | ___ / 3*\/ 332151 | / 3*\/ 332151
3 / 216 + ------------ |\/ 3 *3 / 216 + ------------ ___ | 3 / 216 + ------------ | ___ \/ 3 *3 / 216 + ------------ | 3 / 216 + ------------
5 \/ 8 | \/ 8 5*\/ 3 | 5 \/ 8 | 5*\/ 3 \/ 8 | \/ 8 5
- --------------------------- + ------------------------- + I*|------------------------------- + ---------------------------| + - --------------------------- + ------------------------- + I*|- --------------------------- - -------------------------------| + - ------------------------- + ---------------------------
____________________ 6 | 6 ____________________| ____________________ 6 | ____________________ 6 | 3 ____________________
/ ________ | / ________ | / ________ | / ________ | / ________
/ 3*\/ 332151 | / 3*\/ 332151 | / 3*\/ 332151 | / 3*\/ 332151 | / 3*\/ 332151
8*3 / 216 + ------------ | 8*3 / 216 + ------------ | 8*3 / 216 + ------------ | 8*3 / 216 + ------------ | 4*3 / 216 + ------------
\/ 8 \ \/ 8 / \/ 8 \ \/ 8 / \/ 8
$$\left(- \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{3} + \frac{5}{4 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}\right) + \left(\left(- \frac{5}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} - \frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}\right)\right) + \left(- \frac{5}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} + i \left(\frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6}\right)\right)\right)$$
=
/ ____________________\ / ____________________ \ | / ________ | | / ________ | | ___ / 3*\/ 332151 | | ___ / 3*\/ 332151 | | ___ \/ 3 *3 / 216 + ------------ | |\/ 3 *3 / 216 + ------------ ___ | | 5*\/ 3 \/ 8 | | \/ 8 5*\/ 3 | I*|- --------------------------- - -------------------------------| + I*|------------------------------- + ---------------------------| | ____________________ 6 | | 6 ____________________| | / ________ | | / ________ | | / 3*\/ 332151 | | / 3*\/ 332151 | | 8*3 / 216 + ------------ | | 8*3 / 216 + ------------ | \ \/ 8 / \ \/ 8 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} - \frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}\right) + i \left(\frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6}\right)$$
producto
/ ____________________ / ____________________ \\ / ____________________ / ____________________\\ / ____________________ \ | / ________ | / ________ || | / ________ | / ________ || | / ________ | | / 3*\/ 332151 | ___ / 3*\/ 332151 || | / 3*\/ 332151 | ___ / 3*\/ 332151 || | / 3*\/ 332151 | | 3 / 216 + ------------ |\/ 3 *3 / 216 + ------------ ___ || | 3 / 216 + ------------ | ___ \/ 3 *3 / 216 + ------------ || | 3 / 216 + ------------ | | 5 \/ 8 | \/ 8 5*\/ 3 || | 5 \/ 8 | 5*\/ 3 \/ 8 || | \/ 8 5 | |- --------------------------- + ------------------------- + I*|------------------------------- + ---------------------------||*|- --------------------------- + ------------------------- + I*|- --------------------------- - -------------------------------||*|- ------------------------- + ---------------------------| | ____________________ 6 | 6 ____________________|| | ____________________ 6 | ____________________ 6 || | 3 ____________________| | / ________ | / ________ || | / ________ | / ________ || | / ________ | | / 3*\/ 332151 | / 3*\/ 332151 || | / 3*\/ 332151 | / 3*\/ 332151 || | / 3*\/ 332151 | | 8*3 / 216 + ------------ | 8*3 / 216 + ------------ || | 8*3 / 216 + ------------ | 8*3 / 216 + ------------ || | 4*3 / 216 + ------------ | \ \/ 8 \ \/ 8 // \ \/ 8 \ \/ 8 // \ \/ 8 /
$$\left(- \frac{5}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} + i \left(\frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6}\right)\right) \left(- \frac{5}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} - \frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}\right)\right) \left(- \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{3} + \frac{5}{4 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}\right)$$
=
-16
$$-16$$
-16
Respuesta rápida
[src]
____________________ / ____________________ \
/ ________ | / ________ |
/ 3*\/ 332151 | ___ / 3*\/ 332151 |
3 / 216 + ------------ |\/ 3 *3 / 216 + ------------ ___ |
5 \/ 8 | \/ 8 5*\/ 3 |
x1 = - --------------------------- + ------------------------- + I*|------------------------------- + ---------------------------|
____________________ 6 | 6 ____________________|
/ ________ | / ________ |
/ 3*\/ 332151 | / 3*\/ 332151 |
8*3 / 216 + ------------ | 8*3 / 216 + ------------ |
\/ 8 \ \/ 8 /
$$x_{1} = - \frac{5}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} + i \left(\frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6}\right)$$
____________________ / ____________________\
/ ________ | / ________ |
/ 3*\/ 332151 | ___ / 3*\/ 332151 |
3 / 216 + ------------ | ___ \/ 3 *3 / 216 + ------------ |
5 \/ 8 | 5*\/ 3 \/ 8 |
x2 = - --------------------------- + ------------------------- + I*|- --------------------------- - -------------------------------|
____________________ 6 | ____________________ 6 |
/ ________ | / ________ |
/ 3*\/ 332151 | / 3*\/ 332151 |
8*3 / 216 + ------------ | 8*3 / 216 + ------------ |
\/ 8 \ \/ 8 /
$$x_{2} = - \frac{5}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}} + \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{6} - \frac{5 \sqrt{3}}{8 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}\right)$$
____________________
/ ________
/ 3*\/ 332151
3 / 216 + ------------
\/ 8 5
x3 = - ------------------------- + ---------------------------
3 ____________________
/ ________
/ 3*\/ 332151
4*3 / 216 + ------------
\/ 8
$$x_{3} = - \frac{\sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}{3} + \frac{5}{4 \sqrt[3]{216 + \frac{3 \sqrt{332151}}{8}}}$$
x3 = -(216 + 3*sqrt(332151)/8)^(1/3)/3 + 5/(4*(216 + 3*sqrt(332151)/8)^(1/3))
Respuesta numérica
[src]
x1 = 1.177370322821 - 2.32564026263282*i
x2 = 1.177370322821 + 2.32564026263282*i
x3 = -2.354740645642
x3 = -2.354740645642