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Expresiones idénticas
- 2x³- uno 5x²+36x-1
- 2x³ menos 15x² más 36x menos 1
- 2x³ menos uno 5x² más 36x menos 1
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Expresiones semejantes
- 2x³-15x²-36x-1
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- 2x³+15x²+36x-1
2x³-15x²+36x-1 la ecuación
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Solución
Ha introducido
[src]
3 2 2*x - 15*x + 36*x - 1 = 0
$$\left(36 x + \left(2 x^{3} - 15 x^{2}\right)\right) - 1 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(36 x + \left(2 x^{3} - 15 x^{2}\right)\right) - 1 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{15 x^{2}}{2} + 18 x - \frac{1}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{15}{2}$$
$$q = \frac{c}{a}$$
$$q = 18$$
$$v = \frac{d}{a}$$
$$v = - \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{15}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 18$$
$$x_{1} x_{2} x_{3} = - \frac{1}{2}$$
$$\left(36 x + \left(2 x^{3} - 15 x^{2}\right)\right) - 1 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{15 x^{2}}{2} + 18 x - \frac{1}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{15}{2}$$
$$q = \frac{c}{a}$$
$$q = 18$$
$$v = \frac{d}{a}$$
$$v = - \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{15}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 18$$
$$x_{1} x_{2} x_{3} = - \frac{1}{2}$$
Suma y producto de raíces
[src]
suma
__________________ / __________________\ __________________ / __________________ \ __________________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 1431 81*\/ 78 | ___ / 1431 81*\/ 78 | / 1431 81*\/ 78 | ___ / 1431 81*\/ 78 | / 1431 81*\/ 78
3 / ---- + --------- | ___ \/ 3 *3 / ---- + --------- | 3 / ---- + --------- | \/ 3 *3 / ---- + --------- ___ | 3 / ---- + ---------
5 \/ 8 4 3 | 3*\/ 3 \/ 8 4 | 5 \/ 8 4 3 | \/ 8 4 3*\/ 3 | 5 3 \/ 8 4
- + ----------------------- + ------------------------- + I*|- ------------------------- + -----------------------------| + - + ----------------------- + ------------------------- + I*|- ----------------------------- + -------------------------| + - - ------------------------- - -----------------------
2 6 __________________ | __________________ 6 | 2 6 __________________ | 6 __________________| 2 __________________ 3
/ ____ | / ____ | / ____ | / ____ | / ____
/ 1431 81*\/ 78 | / 1431 81*\/ 78 | / 1431 81*\/ 78 | / 1431 81*\/ 78 | / 1431 81*\/ 78
8*3 / ---- + --------- | 8*3 / ---- + --------- | 8*3 / ---- + --------- | 8*3 / ---- + --------- | 4*3 / ---- + ---------
\/ 8 4 \ \/ 8 4 / \/ 8 4 \ \/ 8 4 / \/ 8 4
$$\left(- \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{3} - \frac{3}{4 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{5}{2}\right) + \left(\left(\frac{3}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{5}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}\right)\right) + \left(\frac{3}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{5}{2} + i \left(- \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6}\right)\right)\right)$$
=
/ __________________\ / __________________ \
| / ____ | | / ____ |
| ___ / 1431 81*\/ 78 | | ___ / 1431 81*\/ 78 |
| ___ \/ 3 *3 / ---- + --------- | | \/ 3 *3 / ---- + --------- ___ |
15 | 3*\/ 3 \/ 8 4 | | \/ 8 4 3*\/ 3 |
-- + I*|- ------------------------- + -----------------------------| + I*|- ----------------------------- + -------------------------|
2 | __________________ 6 | | 6 __________________|
| / ____ | | / ____ |
| / 1431 81*\/ 78 | | / 1431 81*\/ 78 |
| 8*3 / ---- + --------- | | 8*3 / ---- + --------- |
\ \/ 8 4 / \ \/ 8 4 /
$$\frac{15}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}\right) + i \left(- \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6}\right)$$
producto
/ __________________ / __________________\\ / __________________ / __________________ \\ / __________________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 1431 81*\/ 78 | ___ / 1431 81*\/ 78 || | / 1431 81*\/ 78 | ___ / 1431 81*\/ 78 || | / 1431 81*\/ 78 | | 3 / ---- + --------- | ___ \/ 3 *3 / ---- + --------- || | 3 / ---- + --------- | \/ 3 *3 / ---- + --------- ___ || | 3 / ---- + --------- | |5 \/ 8 4 3 | 3*\/ 3 \/ 8 4 || |5 \/ 8 4 3 | \/ 8 4 3*\/ 3 || |5 3 \/ 8 4 | |- + ----------------------- + ------------------------- + I*|- ------------------------- + -----------------------------||*|- + ----------------------- + ------------------------- + I*|- ----------------------------- + -------------------------||*|- - ------------------------- - -----------------------| |2 6 __________________ | __________________ 6 || |2 6 __________________ | 6 __________________|| |2 __________________ 3 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 1431 81*\/ 78 | / 1431 81*\/ 78 || | / 1431 81*\/ 78 | / 1431 81*\/ 78 || | / 1431 81*\/ 78 | | 8*3 / ---- + --------- | 8*3 / ---- + --------- || | 8*3 / ---- + --------- | 8*3 / ---- + --------- || | 4*3 / ---- + --------- | \ \/ 8 4 \ \/ 8 4 // \ \/ 8 4 \ \/ 8 4 // \ \/ 8 4 /
$$\left(\frac{3}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{5}{2} + i \left(- \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6}\right)\right) \left(\frac{3}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{5}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{3} - \frac{3}{4 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{5}{2}\right)$$
=
1/2
$$\frac{1}{2}$$
1/2
Respuesta rápida
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__________________ / __________________\
/ ____ | / ____ |
/ 1431 81*\/ 78 | ___ / 1431 81*\/ 78 |
3 / ---- + --------- | ___ \/ 3 *3 / ---- + --------- |
5 \/ 8 4 3 | 3*\/ 3 \/ 8 4 |
x1 = - + ----------------------- + ------------------------- + I*|- ------------------------- + -----------------------------|
2 6 __________________ | __________________ 6 |
/ ____ | / ____ |
/ 1431 81*\/ 78 | / 1431 81*\/ 78 |
8*3 / ---- + --------- | 8*3 / ---- + --------- |
\/ 8 4 \ \/ 8 4 /
$$x_{1} = \frac{3}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{5}{2} + i \left(- \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6}\right)$$
__________________ / __________________ \
/ ____ | / ____ |
/ 1431 81*\/ 78 | ___ / 1431 81*\/ 78 |
3 / ---- + --------- | \/ 3 *3 / ---- + --------- ___ |
5 \/ 8 4 3 | \/ 8 4 3*\/ 3 |
x2 = - + ----------------------- + ------------------------- + I*|- ----------------------------- + -------------------------|
2 6 __________________ | 6 __________________|
/ ____ | / ____ |
/ 1431 81*\/ 78 | / 1431 81*\/ 78 |
8*3 / ---- + --------- | 8*3 / ---- + --------- |
\/ 8 4 \ \/ 8 4 /
$$x_{2} = \frac{3}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{5}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{6} + \frac{3 \sqrt{3}}{8 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}\right)$$
__________________
/ ____
/ 1431 81*\/ 78
3 / ---- + ---------
5 3 \/ 8 4
x3 = - - ------------------------- - -----------------------
2 __________________ 3
/ ____
/ 1431 81*\/ 78
4*3 / ---- + ---------
\/ 8 4
$$x_{3} = - \frac{\sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}}{3} - \frac{3}{4 \sqrt[3]{\frac{81 \sqrt{78}}{4} + \frac{1431}{8}}} + \frac{5}{2}$$
x3 = -(81*sqrt(78)/4 + 1431/8)^(1/3)/3 - 3/(4*(81*sqrt(78)/4 + 1431/8)^(1/3)) + 5/2
Respuesta numérica
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x1 = 3.73594715920756 + 1.95772728975712*i
x2 = 0.028105681584872
x3 = 3.73594715920756 - 1.95772728975712*i
x3 = 3.73594715920756 - 1.95772728975712*i