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Ecuación 2*x^2+6*x+9=
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- Derivada de:
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x^3-x^2-x+2
- Gráfico de la función y =:
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x^3-x^2-x+2
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Expresiones idénticas
- x^ tres -x^ dos -x+ dos
- x al cubo menos x al cuadrado menos x más 2
- x en el grado tres menos x en el grado dos menos x más dos
- x3-x2-x+2
- x³-x²-x+2
- x en el grado 3-x en el grado 2-x+2
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Expresiones semejantes
- x^3+x^2-x+2
- x^3-x^2+x+2
- x^3-x^2-x-2
x^3-x^2-x+2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - x - x + 2 = 0
$$\left(- x + \left(x^{3} - x^{2}\right)\right) + 2 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 2$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 2$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 2$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 2$$
Respuesta rápida
[src]
________________ / ________________\
/ _____ | / _____ |
/ 43 3*\/ 177 | ___ / 43 3*\/ 177 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 2 2 2 | 2*\/ 3 \/ 2 2 |
x1 = - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------|
3 6 ________________ | ________________ 6 |
/ _____ | / _____ |
/ 43 3*\/ 177 | / 43 3*\/ 177 |
3*3 / -- + --------- | 3*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)$$
________________ / ________________ \
/ _____ | / _____ |
/ 43 3*\/ 177 | ___ / 43 3*\/ 177 |
3 / -- + --------- | \/ 3 *3 / -- + --------- ___ |
1 \/ 2 2 2 | \/ 2 2 2*\/ 3 |
x2 = - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------|
3 6 ________________ | 6 ________________|
/ _____ | / _____ |
/ 43 3*\/ 177 | / 43 3*\/ 177 |
3*3 / -- + --------- | 3*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)$$
________________
/ _____
/ 43 3*\/ 177
3 / -- + ---------
1 4 \/ 2 2
x3 = - - ----------------------- - ---------------------
3 ________________ 3
/ _____
/ 43 3*\/ 177
3*3 / -- + ---------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{3} - \frac{4}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3}$$
x3 = -(3*sqrt(177)/2 + 43/2)^(1/3)/3 - 4/(3*(3*sqrt(177)/2 + 43/2)^(1/3)) + 1/3
Suma y producto de raíces
[src]
suma
________________ / ________________\ ________________ / ________________ \ ________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 43 3*\/ 177 | ___ / 43 3*\/ 177 | / 43 3*\/ 177 | ___ / 43 3*\/ 177 | / 43 3*\/ 177
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- | 3 / -- + --------- | \/ 3 *3 / -- + --------- ___ | 3 / -- + ---------
1 \/ 2 2 2 | 2*\/ 3 \/ 2 2 | 1 \/ 2 2 2 | \/ 2 2 2*\/ 3 | 1 4 \/ 2 2
- + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------| + - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------| + - - ----------------------- - ---------------------
3 6 ________________ | ________________ 6 | 3 6 ________________ | 6 ________________| 3 ________________ 3
/ _____ | / _____ | / _____ | / _____ | / _____
/ 43 3*\/ 177 | / 43 3*\/ 177 | / 43 3*\/ 177 | / 43 3*\/ 177 | / 43 3*\/ 177
3*3 / -- + --------- | 3*3 / -- + --------- | 3*3 / -- + --------- | 3*3 / -- + --------- | 3*3 / -- + ---------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{3} - \frac{4}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3}\right) + \left(\left(\frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)\right) + \left(\frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)\right)\right)$$
=
/ ________________\ / ________________ \
| / _____ | | / _____ |
| ___ / 43 3*\/ 177 | | ___ / 43 3*\/ 177 |
| ___ \/ 3 *3 / -- + --------- | | \/ 3 *3 / -- + --------- ___ |
| 2*\/ 3 \/ 2 2 | | \/ 2 2 2*\/ 3 |
1 + I*|- ----------------------- + ---------------------------| + I*|- --------------------------- + -----------------------|
| ________________ 6 | | 6 ________________|
| / _____ | | / _____ |
| / 43 3*\/ 177 | | / 43 3*\/ 177 |
| 3*3 / -- + --------- | | 3*3 / -- + --------- |
\ \/ 2 2 / \ \/ 2 2 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right) + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________ \\ / ________________\ | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 43 3*\/ 177 | ___ / 43 3*\/ 177 || | / 43 3*\/ 177 | ___ / 43 3*\/ 177 || | / 43 3*\/ 177 | | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- || | 3 / -- + --------- | \/ 3 *3 / -- + --------- ___ || | 3 / -- + --------- | |1 \/ 2 2 2 | 2*\/ 3 \/ 2 2 || |1 \/ 2 2 2 | \/ 2 2 2*\/ 3 || |1 4 \/ 2 2 | |- + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------||*|- + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------||*|- - ----------------------- - ---------------------| |3 6 ________________ | ________________ 6 || |3 6 ________________ | 6 ________________|| |3 ________________ 3 | | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 43 3*\/ 177 | / 43 3*\/ 177 || | / 43 3*\/ 177 | / 43 3*\/ 177 || | / 43 3*\/ 177 | | 3*3 / -- + --------- | 3*3 / -- + --------- || | 3*3 / -- + --------- | 3*3 / -- + --------- || | 3*3 / -- + --------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(\frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)\right) \left(\frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{3} - \frac{4}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{1}{3}\right)$$
=
-2
$$-2$$
-2
Respuesta numérica
[src]
x1 = -1.20556943040059
x2 = 1.1027847152003 - 0.665456951152813*i
x3 = 1.1027847152003 + 0.665456951152813*i
x3 = 1.1027847152003 + 0.665456951152813*i