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Ecuación x^3-3x^2+4=0
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Expresiones idénticas
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- x al cubo menos x al cuadrado más 2 multiplicar por x menos 12 es igual a 0
- x en el grado tres menos x en el grado dos más dos multiplicar por x menos doce es igual a cero
- x3-x2+2*x-12=0
- x³-x²+2*x-12=0
- x en el grado 3-x en el grado 2+2*x-12=0
- x^3-x^2+2x-12=0
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Expresiones semejantes
- x^3-x^2-2*x-12=0
- x^3+x^2+2*x-12=0
- x^3-x^2+2*x+12=0
x^3-x^2+2*x-12=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - x + 2*x - 12 = 0
$$\left(2 x + \left(x^{3} - x^{2}\right)\right) - 12 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -12$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -12$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -12$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -12$$
Suma y producto de raíces
[src]
suma
________________ / ________________\ ________________ / ________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 154 \/ 2649 | ___ / 154 \/ 2649 | / 154 \/ 2649 | ___ / 154 \/ 2649 | ________________
3 / --- + -------- | ___ \/ 3 *3 / --- + -------- | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ | / ______
1 \/ 27 9 5 | 5*\/ 3 \/ 27 9 | 1 \/ 27 9 5 | \/ 27 9 5*\/ 3 | 1 / 154 \/ 2649 5
- - --------------------- + ------------------------ + I*|- ------------------------ - ---------------------------| + - - --------------------- + ------------------------ + I*|--------------------------- + ------------------------| + - + 3 / --- + -------- - -----------------------
3 2 ________________ | ________________ 2 | 3 2 ________________ | 2 ________________| 3 \/ 27 9 ________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 154 \/ 2649 | / 154 \/ 2649 | / 154 \/ 2649 | / 154 \/ 2649 | / 154 \/ 2649
18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 9*3 / --- + --------
\/ 27 9 \ \/ 27 9 / \/ 27 9 \ \/ 27 9 / \/ 27 9
$$\left(- \frac{5}{9 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} + \frac{5}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} - \frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} + \frac{5}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + i \left(\frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________\
| / ______ | | / ______ |
| ___ / 154 \/ 2649 | | ___ / 154 \/ 2649 |
|\/ 3 *3 / --- + -------- ___ | | ___ \/ 3 *3 / --- + -------- |
| \/ 27 9 5*\/ 3 | | 5*\/ 3 \/ 27 9 |
1 + I*|--------------------------- + ------------------------| + I*|- ------------------------ - ---------------------------|
| 2 ________________| | ________________ 2 |
| / ______ | | / ______ |
| / 154 \/ 2649 | | / 154 \/ 2649 |
| 18*3 / --- + -------- | | 18*3 / --- + -------- |
\ \/ 27 9 / \ \/ 27 9 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} - \frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}\right) + i \left(\frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 154 \/ 2649 | ___ / 154 \/ 2649 || | / 154 \/ 2649 | ___ / 154 \/ 2649 || / ________________ \ | 3 / --- + -------- | ___ \/ 3 *3 / --- + -------- || | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ || | / ______ | |1 \/ 27 9 5 | 5*\/ 3 \/ 27 9 || |1 \/ 27 9 5 | \/ 27 9 5*\/ 3 || |1 / 154 \/ 2649 5 | |- - --------------------- + ------------------------ + I*|- ------------------------ - ---------------------------||*|- - --------------------- + ------------------------ + I*|--------------------------- + ------------------------||*|- + 3 / --- + -------- - -----------------------| |3 2 ________________ | ________________ 2 || |3 2 ________________ | 2 ________________|| |3 \/ 27 9 ________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 154 \/ 2649 | / 154 \/ 2649 || | / 154 \/ 2649 | / 154 \/ 2649 || | / 154 \/ 2649 | | 18*3 / --- + -------- | 18*3 / --- + -------- || | 18*3 / --- + -------- | 18*3 / --- + -------- || | 9*3 / --- + -------- | \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 /
$$\left(- \frac{\sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} + \frac{5}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + i \left(\frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} + \frac{5}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} - \frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}\right)\right) \left(- \frac{5}{9 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}\right)$$
=
12
$$12$$
12
Respuesta rápida
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________________ / ________________\
/ ______ | / ______ |
/ 154 \/ 2649 | ___ / 154 \/ 2649 |
3 / --- + -------- | ___ \/ 3 *3 / --- + -------- |
1 \/ 27 9 5 | 5*\/ 3 \/ 27 9 |
x1 = - - --------------------- + ------------------------ + I*|- ------------------------ - ---------------------------|
3 2 ________________ | ________________ 2 |
/ ______ | / ______ |
/ 154 \/ 2649 | / 154 \/ 2649 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 27 9 \ \/ 27 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} + \frac{5}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} - \frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}\right)$$
________________ / ________________ \
/ ______ | / ______ |
/ 154 \/ 2649 | ___ / 154 \/ 2649 |
3 / --- + -------- |\/ 3 *3 / --- + -------- ___ |
1 \/ 27 9 5 | \/ 27 9 5*\/ 3 |
x2 = - - --------------------- + ------------------------ + I*|--------------------------- + ------------------------|
3 2 ________________ | 2 ________________|
/ ______ | / ______ |
/ 154 \/ 2649 | / 154 \/ 2649 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 27 9 \ \/ 27 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2} + \frac{5}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + i \left(\frac{5 \sqrt{3}}{18 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}}{2}\right)$$
________________
/ ______
1 / 154 \/ 2649 5
x3 = - + 3 / --- + -------- - -----------------------
3 \/ 27 9 ________________
/ ______
/ 154 \/ 2649
9*3 / --- + --------
\/ 27 9
$$x_{3} = - \frac{5}{9 \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}} + \frac{1}{3} + \sqrt[3]{\frac{154}{27} + \frac{\sqrt{2649}}{9}}$$
x3 = -5/(9*(154/27 + sqrt(2649)/9)^(1/3)) + 1/3 + (154/27 + sqrt(2649)/9)^(1/3)
Respuesta numérica
[src]
x1 = -0.669370269911302 - 2.16400328431959*i
x2 = 2.3387405398226
x3 = -0.669370269911302 + 2.16400328431959*i
x3 = -0.669370269911302 + 2.16400328431959*i