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Expresiones idénticas
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Expresiones semejantes
- -x^3-4*x^2-4*x+192=0
- x^3+4*x^2-4*x+192=0
- -x^3+4*x^2+4*x+192=0
- -x^3+4*x^2-4*x-192=0
-x^3+4*x^2-4*x+192=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 - x + 4*x - 4*x + 192 = 0
$$\left(- 4 x + \left(- x^{3} + 4 x^{2}\right)\right) + 192 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 4 x + \left(- x^{3} + 4 x^{2}\right)\right) + 192 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x^{2} + 4 x - 192 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 4$$
$$v = \frac{d}{a}$$
$$v = -192$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 4$$
$$x_{1} x_{2} x_{3} = -192$$
$$\left(- 4 x + \left(- x^{3} + 4 x^{2}\right)\right) + 192 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x^{2} + 4 x - 192 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 4$$
$$v = \frac{d}{a}$$
$$v = -192$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 4$$
$$x_{1} x_{2} x_{3} = -192$$
Suma y producto de raíces
[src]
suma
_________________ / _________________ \ _________________ / _________________ \ _________________
/ _____ | / _____ ___ | / _____ | / _____ ___ | / _____
4 / 323 2*\/ 322 1 | ___ / 323 2*\/ 322 \/ 3 | 4 / 323 2*\/ 322 1 | ___ / 323 2*\/ 322 \/ 3 | 4 / 323 2*\/ 322 2
- - 3 / --- + --------- - ------------------------ + I*|- \/ 3 *3 / --- + --------- + ------------------------| + - - 3 / --- + --------- - ------------------------ + I*|\/ 3 *3 / --- + --------- - ------------------------| + - + 2*3 / --- + --------- + ------------------------
3 \/ 27 3 _________________ | \/ 27 3 _________________| 3 \/ 27 3 _________________ | \/ 27 3 _________________| 3 \/ 27 3 _________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 323 2*\/ 322 | / 323 2*\/ 322 | / 323 2*\/ 322 | / 323 2*\/ 322 | / 323 2*\/ 322
9*3 / --- + --------- | 9*3 / --- + --------- | 9*3 / --- + --------- | 9*3 / --- + --------- | 9*3 / --- + ---------
\/ 27 3 \ \/ 27 3 / \/ 27 3 \ \/ 27 3 / \/ 27 3
$$\left(\frac{2}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + 2 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}\right) + \left(\left(- \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} - \frac{1}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + i \left(- \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} + \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}}\right)\right) + \left(- \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} - \frac{1}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}\right)\right)\right)$$
=
/ _________________ \ / _________________ \
| / _____ ___ | | / _____ ___ |
| ___ / 323 2*\/ 322 \/ 3 | | ___ / 323 2*\/ 322 \/ 3 |
4 + I*|\/ 3 *3 / --- + --------- - ------------------------| + I*|- \/ 3 *3 / --- + --------- + ------------------------|
| \/ 27 3 _________________| | \/ 27 3 _________________|
| / _____ | | / _____ |
| / 323 2*\/ 322 | | / 323 2*\/ 322 |
| 9*3 / --- + --------- | | 9*3 / --- + --------- |
\ \/ 27 3 / \ \/ 27 3 /
$$4 + i \left(- \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} + \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}}\right) + i \left(- \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}\right)$$
producto
/ _________________ / _________________ \\ / _________________ / _________________ \\ / _________________ \ | / _____ | / _____ ___ || | / _____ | / _____ ___ || | / _____ | |4 / 323 2*\/ 322 1 | ___ / 323 2*\/ 322 \/ 3 || |4 / 323 2*\/ 322 1 | ___ / 323 2*\/ 322 \/ 3 || |4 / 323 2*\/ 322 2 | |- - 3 / --- + --------- - ------------------------ + I*|- \/ 3 *3 / --- + --------- + ------------------------||*|- - 3 / --- + --------- - ------------------------ + I*|\/ 3 *3 / --- + --------- - ------------------------||*|- + 2*3 / --- + --------- + ------------------------| |3 \/ 27 3 _________________ | \/ 27 3 _________________|| |3 \/ 27 3 _________________ | \/ 27 3 _________________|| |3 \/ 27 3 _________________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 323 2*\/ 322 | / 323 2*\/ 322 || | / 323 2*\/ 322 | / 323 2*\/ 322 || | / 323 2*\/ 322 | | 9*3 / --- + --------- | 9*3 / --- + --------- || | 9*3 / --- + --------- | 9*3 / --- + --------- || | 9*3 / --- + --------- | \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 /
$$\left(- \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} - \frac{1}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}\right)\right) \left(- \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} - \frac{1}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + i \left(- \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} + \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}}\right)\right) \left(\frac{2}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + 2 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}\right)$$
=
192
$$192$$
192
Respuesta rápida
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_________________ / _________________ \
/ _____ | / _____ ___ |
4 / 323 2*\/ 322 1 | ___ / 323 2*\/ 322 \/ 3 |
x1 = - - 3 / --- + --------- - ------------------------ + I*|- \/ 3 *3 / --- + --------- + ------------------------|
3 \/ 27 3 _________________ | \/ 27 3 _________________|
/ _____ | / _____ |
/ 323 2*\/ 322 | / 323 2*\/ 322 |
9*3 / --- + --------- | 9*3 / --- + --------- |
\/ 27 3 \ \/ 27 3 /
$$x_{1} = - \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} - \frac{1}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + i \left(- \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} + \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}}\right)$$
_________________ / _________________ \
/ _____ | / _____ ___ |
4 / 323 2*\/ 322 1 | ___ / 323 2*\/ 322 \/ 3 |
x2 = - - 3 / --- + --------- - ------------------------ + I*|\/ 3 *3 / --- + --------- - ------------------------|
3 \/ 27 3 _________________ | \/ 27 3 _________________|
/ _____ | / _____ |
/ 323 2*\/ 322 | / 323 2*\/ 322 |
9*3 / --- + --------- | 9*3 / --- + --------- |
\/ 27 3 \ \/ 27 3 /
$$x_{2} = - \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}} - \frac{1}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \sqrt{3} \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}\right)$$
_________________
/ _____
4 / 323 2*\/ 322 2
x3 = - + 2*3 / --- + --------- + ------------------------
3 \/ 27 3 _________________
/ _____
/ 323 2*\/ 322
9*3 / --- + ---------
\/ 27 3
$$x_{3} = \frac{2}{9 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}} + \frac{4}{3} + 2 \sqrt[3]{\frac{2 \sqrt{322}}{3} + \frac{323}{27}}$$
x3 = 2/(9*(2*sqrt(322)/3 + 323/27)^(1/3)) + 4/3 + 2*(2*sqrt(322)/3 + 323/27)^(1/3)
Respuesta numérica
[src]
x1 = -1.58675266864455 - 4.92416220727021*i
x2 = -1.58675266864455 + 4.92416220727021*i
x3 = 7.1735053372891
x3 = 7.1735053372891