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Expresiones idénticas
- x^ tres + dos x^2+9x- dieciocho = cero
- x al cubo más 2x al cuadrado más 9x menos 18 es igual a 0
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- x3+2x2+9x-18=0
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Expresiones semejantes
- x^3+2x^2+9x+18=0
- x^3+2x^2-9x-18=0
- x^3-2x^2+9x-18=0
x^3+2x^2+9x-18=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x + 2*x + 9*x - 18 = 0
$$\left(9 x + \left(x^{3} + 2 x^{2}\right)\right) - 18 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = 9$$
$$v = \frac{d}{a}$$
$$v = -18$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 9$$
$$x_{1} x_{2} x_{3} = -18$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = 9$$
$$v = \frac{d}{a}$$
$$v = -18$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 9$$
$$x_{1} x_{2} x_{3} = -18$$
Respuesta rápida
[src]
________________ / ________________\
/ ______ | / ______ |
/ 316 \/ 1383 | ___ / 316 \/ 1383 |
3 / --- + -------- | ___ \/ 3 *3 / --- + -------- |
2 \/ 27 3 23 | 23*\/ 3 \/ 27 3 |
x1 = - - - --------------------- + ------------------------ + I*|- ------------------------ - ---------------------------|
3 2 ________________ | ________________ 2 |
/ ______ | / ______ |
/ 316 \/ 1383 | / 316 \/ 1383 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 27 3 \ \/ 27 3 /
$$x_{1} = - \frac{\sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{2}{3} + \frac{23}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}\right)$$
________________ / ________________ \
/ ______ | / ______ |
/ 316 \/ 1383 | ___ / 316 \/ 1383 |
3 / --- + -------- |\/ 3 *3 / --- + -------- ___ |
2 \/ 27 3 23 | \/ 27 3 23*\/ 3 |
x2 = - - - --------------------- + ------------------------ + I*|--------------------------- + ------------------------|
3 2 ________________ | 2 ________________|
/ ______ | / ______ |
/ 316 \/ 1383 | / 316 \/ 1383 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 27 3 \ \/ 27 3 /
$$x_{2} = - \frac{\sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{2}{3} + \frac{23}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + i \left(\frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2}\right)$$
________________
/ ______
2 / 316 \/ 1383 23
x3 = - - + 3 / --- + -------- - -----------------------
3 \/ 27 3 ________________
/ ______
/ 316 \/ 1383
9*3 / --- + --------
\/ 27 3
$$x_{3} = - \frac{23}{9 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} - \frac{2}{3} + \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}$$
x3 = -23/(9*(316/27 + sqrt(1383)/3)^(1/3)) - 2/3 + (316/27 + sqrt(1383)/3)^(1/3)
Suma y producto de raíces
[src]
suma
________________ / ________________\ ________________ / ________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 316 \/ 1383 | ___ / 316 \/ 1383 | / 316 \/ 1383 | ___ / 316 \/ 1383 | ________________
3 / --- + -------- | ___ \/ 3 *3 / --- + -------- | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ | / ______
2 \/ 27 3 23 | 23*\/ 3 \/ 27 3 | 2 \/ 27 3 23 | \/ 27 3 23*\/ 3 | 2 / 316 \/ 1383 23
- - - --------------------- + ------------------------ + I*|- ------------------------ - ---------------------------| + - - - --------------------- + ------------------------ + I*|--------------------------- + ------------------------| + - - + 3 / --- + -------- - -----------------------
3 2 ________________ | ________________ 2 | 3 2 ________________ | 2 ________________| 3 \/ 27 3 ________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 316 \/ 1383 | / 316 \/ 1383 | / 316 \/ 1383 | / 316 \/ 1383 | / 316 \/ 1383
18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 9*3 / --- + --------
\/ 27 3 \ \/ 27 3 / \/ 27 3 \ \/ 27 3 / \/ 27 3
$$\left(- \frac{23}{9 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} - \frac{2}{3} + \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{2}{3} + \frac{23}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{2}{3} + \frac{23}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + i \left(\frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________\
| / ______ | | / ______ |
| ___ / 316 \/ 1383 | | ___ / 316 \/ 1383 |
|\/ 3 *3 / --- + -------- ___ | | ___ \/ 3 *3 / --- + -------- |
| \/ 27 3 23*\/ 3 | | 23*\/ 3 \/ 27 3 |
-2 + I*|--------------------------- + ------------------------| + I*|- ------------------------ - ---------------------------|
| 2 ________________| | ________________ 2 |
| / ______ | | / ______ |
| / 316 \/ 1383 | | / 316 \/ 1383 |
| 18*3 / --- + -------- | | 18*3 / --- + -------- |
\ \/ 27 3 / \ \/ 27 3 /
$$-2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}\right) + i \left(\frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 316 \/ 1383 | ___ / 316 \/ 1383 || | / 316 \/ 1383 | ___ / 316 \/ 1383 || / ________________ \ | 3 / --- + -------- | ___ \/ 3 *3 / --- + -------- || | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ || | / ______ | | 2 \/ 27 3 23 | 23*\/ 3 \/ 27 3 || | 2 \/ 27 3 23 | \/ 27 3 23*\/ 3 || | 2 / 316 \/ 1383 23 | |- - - --------------------- + ------------------------ + I*|- ------------------------ - ---------------------------||*|- - - --------------------- + ------------------------ + I*|--------------------------- + ------------------------||*|- - + 3 / --- + -------- - -----------------------| | 3 2 ________________ | ________________ 2 || | 3 2 ________________ | 2 ________________|| | 3 \/ 27 3 ________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 316 \/ 1383 | / 316 \/ 1383 || | / 316 \/ 1383 | / 316 \/ 1383 || | / 316 \/ 1383 | | 18*3 / --- + -------- | 18*3 / --- + -------- || | 18*3 / --- + -------- | 18*3 / --- + -------- || | 9*3 / --- + -------- | \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 /
$$\left(- \frac{\sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{2}{3} + \frac{23}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + i \left(\frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{2}{3} + \frac{23}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}{2} - \frac{23 \sqrt{3}}{18 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}}\right)\right) \left(- \frac{23}{9 \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}} - \frac{2}{3} + \sqrt[3]{\frac{316}{27} + \frac{\sqrt{1383}}{3}}\right)$$
=
18
$$18$$
18
Respuesta numérica
[src]
x1 = -1.66854774542557 + 3.26771537233461*i
x2 = -1.66854774542557 - 3.26771537233461*i
x3 = 1.33709549085113
x3 = 1.33709549085113