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Ecuación x^2-5x=14
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Ecuación x+2=-x+14
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Expresiones idénticas
- x^ tres + cuatro x^ dos -x+4= cero
- x al cubo más 4x al cuadrado menos x más 4 es igual a 0
- x en el grado tres más cuatro x en el grado dos menos x más 4 es igual a cero
- x3+4x2-x+4=0
- x³+4x²-x+4=0
- x en el grado 3+4x en el grado 2-x+4=0
- x^3+4x^2-x+4=O
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Expresiones semejantes
- x^3+4x^2+x+4=0
- x^3-4x^2-x+4=0
- x^3+4x^2-x-4=0
x^3+4x^2-x+4=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x + 4*x - x + 4 = 0
$$\left(- x + \left(x^{3} + 4 x^{2}\right)\right) + 4 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 4$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 4$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 4$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 4$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 4$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 4$$
Respuesta rápida
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__________________ / __________________\
3 / ______ | ___ ___ 3 / ______ |
4 \/ 136 + 3*\/ 1293 19 | 19*\/ 3 \/ 3 *\/ 136 + 3*\/ 1293 |
x1 = - - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------|
3 6 __________________ | __________________ 6 |
3 / ______ | 3 / ______ |
6*\/ 136 + 3*\/ 1293 \ 6*\/ 136 + 3*\/ 1293 /
$$x_{1} = - \frac{4}{3} + \frac{19}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{6} + i \left(- \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6}\right)$$
__________________ / __________________ \
3 / ______ | ___ 3 / ______ ___ |
4 \/ 136 + 3*\/ 1293 19 | \/ 3 *\/ 136 + 3*\/ 1293 19*\/ 3 |
x2 = - - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------|
3 6 __________________ | 6 __________________|
3 / ______ | 3 / ______ |
6*\/ 136 + 3*\/ 1293 \ 6*\/ 136 + 3*\/ 1293 /
$$x_{2} = - \frac{4}{3} + \frac{19}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6} + \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}}\right)$$
__________________
3 / ______
4 19 \/ 136 + 3*\/ 1293
x3 = - - - ----------------------- - ---------------------
3 __________________ 3
3 / ______
3*\/ 136 + 3*\/ 1293
$$x_{3} = - \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{3} - \frac{4}{3} - \frac{19}{3 \sqrt[3]{3 \sqrt{1293} + 136}}$$
x3 = -(3*sqrt(1293) + 136)^(1/3)/3 - 4/3 - 19/(3*(3*sqrt(1293) + 136)^(1/3))
Suma y producto de raíces
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suma
__________________ / __________________\ __________________ / __________________ \ __________________
3 / ______ | ___ ___ 3 / ______ | 3 / ______ | ___ 3 / ______ ___ | 3 / ______
4 \/ 136 + 3*\/ 1293 19 | 19*\/ 3 \/ 3 *\/ 136 + 3*\/ 1293 | 4 \/ 136 + 3*\/ 1293 19 | \/ 3 *\/ 136 + 3*\/ 1293 19*\/ 3 | 4 19 \/ 136 + 3*\/ 1293
- - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------| + - - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------| + - - - ----------------------- - ---------------------
3 6 __________________ | __________________ 6 | 3 6 __________________ | 6 __________________| 3 __________________ 3
3 / ______ | 3 / ______ | 3 / ______ | 3 / ______ | 3 / ______
6*\/ 136 + 3*\/ 1293 \ 6*\/ 136 + 3*\/ 1293 / 6*\/ 136 + 3*\/ 1293 \ 6*\/ 136 + 3*\/ 1293 / 3*\/ 136 + 3*\/ 1293
$$\left(- \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{3} - \frac{4}{3} - \frac{19}{3 \sqrt[3]{3 \sqrt{1293} + 136}}\right) + \left(\left(- \frac{4}{3} + \frac{19}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6} + \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}}\right)\right) + \left(- \frac{4}{3} + \frac{19}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{6} + i \left(- \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6}\right)\right)\right)$$
=
/ __________________\ / __________________ \
| ___ ___ 3 / ______ | | ___ 3 / ______ ___ |
| 19*\/ 3 \/ 3 *\/ 136 + 3*\/ 1293 | | \/ 3 *\/ 136 + 3*\/ 1293 19*\/ 3 |
-4 + I*|- ----------------------- + ---------------------------| + I*|- --------------------------- + -----------------------|
| __________________ 6 | | 6 __________________|
| 3 / ______ | | 3 / ______ |
\ 6*\/ 136 + 3*\/ 1293 / \ 6*\/ 136 + 3*\/ 1293 /
$$-4 + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6} + \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}}\right) + i \left(- \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6}\right)$$
producto
/ __________________ / __________________\\ / __________________ / __________________ \\ / __________________\ | 3 / ______ | ___ ___ 3 / ______ || | 3 / ______ | ___ 3 / ______ ___ || | 3 / ______ | | 4 \/ 136 + 3*\/ 1293 19 | 19*\/ 3 \/ 3 *\/ 136 + 3*\/ 1293 || | 4 \/ 136 + 3*\/ 1293 19 | \/ 3 *\/ 136 + 3*\/ 1293 19*\/ 3 || | 4 19 \/ 136 + 3*\/ 1293 | |- - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------||*|- - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------||*|- - - ----------------------- - ---------------------| | 3 6 __________________ | __________________ 6 || | 3 6 __________________ | 6 __________________|| | 3 __________________ 3 | | 3 / ______ | 3 / ______ || | 3 / ______ | 3 / ______ || | 3 / ______ | \ 6*\/ 136 + 3*\/ 1293 \ 6*\/ 136 + 3*\/ 1293 // \ 6*\/ 136 + 3*\/ 1293 \ 6*\/ 136 + 3*\/ 1293 // \ 3*\/ 136 + 3*\/ 1293 /
$$\left(- \frac{4}{3} + \frac{19}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{6} + i \left(- \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6}\right)\right) \left(- \frac{4}{3} + \frac{19}{6 \sqrt[3]{3 \sqrt{1293} + 136}} + \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1293} + 136}}{6} + \frac{19 \sqrt{3}}{6 \sqrt[3]{3 \sqrt{1293} + 136}}\right)\right) \left(- \frac{\sqrt[3]{3 \sqrt{1293} + 136}}{3} - \frac{4}{3} - \frac{19}{3 \sqrt[3]{3 \sqrt{1293} + 136}}\right)$$
=
-4
$$-4$$
-4
Respuesta numérica
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x1 = -4.42961183346797
x2 = 0.214805916733986 - 0.925673851560946*i
x3 = 0.214805916733986 + 0.925673851560946*i
x3 = 0.214805916733986 + 0.925673851560946*i