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Expresiones semejantes
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3+5x-(4x*(-4x)*(-4x)) la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- - 4 x - 4 x 4 x + \left(5 x + 3\right) = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{5 x}{64} - \frac{3}{64} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{64}$$
$$v = \frac{d}{a}$$
$$v = - \frac{3}{64}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{64}$$
$$x_{1} x_{2} x_{3} = - \frac{3}{64}$$
$$- - 4 x - 4 x 4 x + \left(5 x + 3\right) = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{5 x}{64} - \frac{3}{64} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{64}$$
$$v = \frac{d}{a}$$
$$v = - \frac{3}{64}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{64}$$
$$x_{1} x_{2} x_{3} = - \frac{3}{64}$$
Suma y producto de raíces
[src]
suma
_________________ / _________________ \ _________________ / _________________ \
/ _______ | / _______ | / _______ | / _______ |
/ 3 \/ 11289 | ___ / 3 \/ 11289 | / 3 \/ 11289 | ___ / 3 \/ 11289 | _________________
3 / --- + --------- | \/ 3 *3 / --- + --------- ___ | 3 / --- + --------- |\/ 3 *3 / --- + --------- ___ | / _______
5 \/ 128 4608 | \/ 128 4608 5*\/ 3 | 5 \/ 128 4608 | \/ 128 4608 5*\/ 3 | / 3 \/ 11289 5
- -------------------------- - ---------------------- + I*|- ---------------------------- + --------------------------| + - -------------------------- - ---------------------- + I*|---------------------------- - --------------------------| + 3 / --- + --------- + --------------------------
_________________ 2 | 2 _________________| _________________ 2 | 2 _________________| \/ 128 4608 _________________
/ _______ | / _______ | / _______ | / _______ | / _______
/ 3 \/ 11289 | / 3 \/ 11289 | / 3 \/ 11289 | / 3 \/ 11289 | / 3 \/ 11289
384*3 / --- + --------- | 384*3 / --- + --------- | 384*3 / --- + --------- | 384*3 / --- + --------- | 192*3 / --- + ---------
\/ 128 4608 \ \/ 128 4608 / \/ 128 4608 \ \/ 128 4608 / \/ 128 4608
$$\left(\frac{5}{192 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} - \frac{5}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} + \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} - \frac{5}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + i \left(- \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2}\right)\right)\right)$$
=
/ _________________ \ / _________________ \ | / _______ | | / _______ | | ___ / 3 \/ 11289 | | ___ / 3 \/ 11289 | |\/ 3 *3 / --- + --------- ___ | | \/ 3 *3 / --- + --------- ___ | | \/ 128 4608 5*\/ 3 | | \/ 128 4608 5*\/ 3 | I*|---------------------------- - --------------------------| + I*|- ---------------------------- + --------------------------| | 2 _________________| | 2 _________________| | / _______ | | / _______ | | / 3 \/ 11289 | | / 3 \/ 11289 | | 384*3 / --- + --------- | | 384*3 / --- + --------- | \ \/ 128 4608 / \ \/ 128 4608 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} + \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}\right) + i \left(- \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2}\right)$$
producto
/ _________________ / _________________ \\ / _________________ / _________________ \\ | / _______ | / _______ || | / _______ | / _______ || | / 3 \/ 11289 | ___ / 3 \/ 11289 || | / 3 \/ 11289 | ___ / 3 \/ 11289 || / _________________ \ | 3 / --- + --------- | \/ 3 *3 / --- + --------- ___ || | 3 / --- + --------- |\/ 3 *3 / --- + --------- ___ || | / _______ | | 5 \/ 128 4608 | \/ 128 4608 5*\/ 3 || | 5 \/ 128 4608 | \/ 128 4608 5*\/ 3 || | / 3 \/ 11289 5 | |- -------------------------- - ---------------------- + I*|- ---------------------------- + --------------------------||*|- -------------------------- - ---------------------- + I*|---------------------------- - --------------------------||*|3 / --- + --------- + --------------------------| | _________________ 2 | 2 _________________|| | _________________ 2 | 2 _________________|| |\/ 128 4608 _________________| | / _______ | / _______ || | / _______ | / _______ || | / _______ | | / 3 \/ 11289 | / 3 \/ 11289 || | / 3 \/ 11289 | / 3 \/ 11289 || | / 3 \/ 11289 | | 384*3 / --- + --------- | 384*3 / --- + --------- || | 384*3 / --- + --------- | 384*3 / --- + --------- || | 192*3 / --- + --------- | \ \/ 128 4608 \ \/ 128 4608 // \ \/ 128 4608 \ \/ 128 4608 // \ \/ 128 4608 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} - \frac{5}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + i \left(- \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} - \frac{5}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} + \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}\right)\right) \left(\frac{5}{192 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}\right)$$
=
3/64
$$\frac{3}{64}$$
3/64
Respuesta rápida
[src]
_________________ / _________________ \
/ _______ | / _______ |
/ 3 \/ 11289 | ___ / 3 \/ 11289 |
3 / --- + --------- | \/ 3 *3 / --- + --------- ___ |
5 \/ 128 4608 | \/ 128 4608 5*\/ 3 |
x1 = - -------------------------- - ---------------------- + I*|- ---------------------------- + --------------------------|
_________________ 2 | 2 _________________|
/ _______ | / _______ |
/ 3 \/ 11289 | / 3 \/ 11289 |
384*3 / --- + --------- | 384*3 / --- + --------- |
\/ 128 4608 \ \/ 128 4608 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} - \frac{5}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} + \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}\right)$$
_________________ / _________________ \
/ _______ | / _______ |
/ 3 \/ 11289 | ___ / 3 \/ 11289 |
3 / --- + --------- |\/ 3 *3 / --- + --------- ___ |
5 \/ 128 4608 | \/ 128 4608 5*\/ 3 |
x2 = - -------------------------- - ---------------------- + I*|---------------------------- - --------------------------|
_________________ 2 | 2 _________________|
/ _______ | / _______ |
/ 3 \/ 11289 | / 3 \/ 11289 |
384*3 / --- + --------- | 384*3 / --- + --------- |
\/ 128 4608 \ \/ 128 4608 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2} - \frac{5}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + i \left(- \frac{5 \sqrt{3}}{384 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}}{2}\right)$$
_________________
/ _______
/ 3 \/ 11289 5
x3 = 3 / --- + --------- + --------------------------
\/ 128 4608 _________________
/ _______
/ 3 \/ 11289
192*3 / --- + ---------
\/ 128 4608
$$x_{3} = \frac{5}{192 \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}} + \sqrt[3]{\frac{\sqrt{11289}}{4608} + \frac{3}{128}}$$
x3 = 5/(192*(sqrt(11289)/4608 + 3/128)^(1/3)) + (sqrt(11289)/4608 + 3/128)^(1/3)
Respuesta numérica
[src]
x1 = 0.432007123934819
x2 = -0.216003561967409 - 0.248691809973359*i
x3 = -0.216003561967409 + 0.248691809973359*i
x3 = -0.216003561967409 + 0.248691809973359*i