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Expresiones idénticas
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- (4*x^3)+(2*x)+1=O
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Expresiones semejantes
- (4*x^3)+(2*x)-1=0
- (4*x^3)-(2*x)+1=0
(4*x^3)+(2*x)+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(4 x^{3} + 2 x\right) + 1 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{x}{2} + \frac{1}{4} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{1}{4}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{2}$$
$$x_{1} x_{2} x_{3} = \frac{1}{4}$$
$$\left(4 x^{3} + 2 x\right) + 1 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{x}{2} + \frac{1}{4} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{1}{4}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{2}$$
$$x_{1} x_{2} x_{3} = \frac{1}{4}$$
Suma y producto de raíces
[src]
suma
________________ / ________________\ ________________ / ________________\ ________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 27 3*\/ 105 | ___ / 27 3*\/ 105 | / 27 3*\/ 105 | ___ / 27 3*\/ 105 | / 27 3*\/ 105
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- | 3 / -- + ---------
1 \/ 8 8 | \/ 3 \/ 8 8 | 1 \/ 8 8 | \/ 3 \/ 8 8 | 1 \/ 8 8
- ----------------------- + --------------------- + I*|----------------------- + ---------------------------| + - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------| + ----------------------- - ---------------------
________________ 6 | ________________ 6 | ________________ 6 | ________________ 6 | ________________ 3
/ _____ | / _____ | / _____ | / _____ | / _____
/ 27 3*\/ 105 | / 27 3*\/ 105 | / 27 3*\/ 105 | / 27 3*\/ 105 | / 27 3*\/ 105
4*3 / -- + --------- |4*3 / -- + --------- | 4*3 / -- + --------- | 4*3 / -- + --------- | 2*3 / -- + ---------
\/ 8 8 \ \/ 8 8 / \/ 8 8 \ \/ 8 8 / \/ 8 8
$$\left(- \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{3} + \frac{1}{2 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right) + \left(\left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)\right) + \left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)\right)\right)$$
=
/ ________________\ / ________________\ | / _____ | | / _____ | | ___ / 27 3*\/ 105 | | ___ / 27 3*\/ 105 | | ___ \/ 3 *3 / -- + --------- | | ___ \/ 3 *3 / -- + --------- | | \/ 3 \/ 8 8 | | \/ 3 \/ 8 8 | I*|- ----------------------- - ---------------------------| + I*|----------------------- + ---------------------------| | ________________ 6 | | ________________ 6 | | / _____ | | / _____ | | / 27 3*\/ 105 | | / 27 3*\/ 105 | | 4*3 / -- + --------- | |4*3 / -- + --------- | \ \/ 8 8 / \ \/ 8 8 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right) + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________\\ / ________________\ | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 27 3*\/ 105 | ___ / 27 3*\/ 105 || | / 27 3*\/ 105 | ___ / 27 3*\/ 105 || | / 27 3*\/ 105 | | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- || | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- || | 3 / -- + --------- | | 1 \/ 8 8 | \/ 3 \/ 8 8 || | 1 \/ 8 8 | \/ 3 \/ 8 8 || | 1 \/ 8 8 | |- ----------------------- + --------------------- + I*|----------------------- + ---------------------------||*|- ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------||*|----------------------- - ---------------------| | ________________ 6 | ________________ 6 || | ________________ 6 | ________________ 6 || | ________________ 3 | | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 27 3*\/ 105 | / 27 3*\/ 105 || | / 27 3*\/ 105 | / 27 3*\/ 105 || | / 27 3*\/ 105 | | 4*3 / -- + --------- |4*3 / -- + --------- || | 4*3 / -- + --------- | 4*3 / -- + --------- || |2*3 / -- + --------- | \ \/ 8 8 \ \/ 8 8 // \ \/ 8 8 \ \/ 8 8 // \ \/ 8 8 /
$$\left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)\right) \left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{3} + \frac{1}{2 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)$$
=
-1/4
$$- \frac{1}{4}$$
-1/4
Respuesta rápida
[src]
________________ / ________________\
/ _____ | / _____ |
/ 27 3*\/ 105 | ___ / 27 3*\/ 105 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 8 8 | \/ 3 \/ 8 8 |
x1 = - ----------------------- + --------------------- + I*|----------------------- + ---------------------------|
________________ 6 | ________________ 6 |
/ _____ | / _____ |
/ 27 3*\/ 105 | / 27 3*\/ 105 |
4*3 / -- + --------- |4*3 / -- + --------- |
\/ 8 8 \ \/ 8 8 /
$$x_{1} = - \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)$$
________________ / ________________\
/ _____ | / _____ |
/ 27 3*\/ 105 | ___ / 27 3*\/ 105 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 8 8 | \/ 3 \/ 8 8 |
x2 = - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------|
________________ 6 | ________________ 6 |
/ _____ | / _____ |
/ 27 3*\/ 105 | / 27 3*\/ 105 |
4*3 / -- + --------- | 4*3 / -- + --------- |
\/ 8 8 \ \/ 8 8 /
$$x_{2} = - \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)$$
________________
/ _____
/ 27 3*\/ 105
3 / -- + ---------
1 \/ 8 8
x3 = ----------------------- - ---------------------
________________ 3
/ _____
/ 27 3*\/ 105
2*3 / -- + ---------
\/ 8 8
$$x_{3} = - \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{3} + \frac{1}{2 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}$$
x3 = -(27/8 + 3*sqrt(105)/8)^(1/3)/3 + 1/(2*(27/8 + 3*sqrt(105)/8)^(1/3))
Respuesta numérica
[src]
x1 = -0.385458498529624
x2 = 0.192729249264812 - 0.781942255263478*i
x3 = 0.192729249264812 + 0.781942255263478*i
x3 = 0.192729249264812 + 0.781942255263478*i