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-x^4+2*x-10=0 la ecuación
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Solución
Suma y producto de raíces
[src]
suma
_______________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________ _______________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________
/ _________________ / _________________ / _________________ / _________________
/ / _______ / / _______ / / _______ / / _______
/ / 1 I*\/ 47919 4 20 / 4 / 1 I*\/ 47919 20 / / 1 I*\/ 47919 4 20 / 4 / 1 I*\/ 47919 20
I* / 2*3 / - + ----------- + --------------------------------------------------------------- + ------------------------ I* / - --------------------------------------------------------------- + 2*3 / - + ----------- + ------------------------ I* / 2*3 / - + ----------- + --------------------------------------------------------------- + ------------------------ I* / - --------------------------------------------------------------- + 2*3 / - + ----------- + ------------------------
_____________________________________________________ / \/ 4 36 _____________________________________________________ _________________ _____________________________________________________ / _____________________________________________________ \/ 4 36 _________________ _____________________________________________________ / \/ 4 36 _____________________________________________________ _________________ _____________________________________________________ / _____________________________________________________ \/ 4 36 _________________
/ _________________ / / _________________ / _______ / _________________ / / _________________ / _______ / _________________ / / _________________ / _______ / _________________ / / _________________ / _______
/ / _______ / / / _______ / 1 I*\/ 47919 / / _______ / / / _______ / 1 I*\/ 47919 / / _______ / / / _______ / 1 I*\/ 47919 / / _______ / / / _______ / 1 I*\/ 47919
/ / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + -----------
/ 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36
/ \/ 4 36 _________________ / / \/ 4 36 _________________ / \/ 4 36 _________________ / / \/ 4 36 _________________ / \/ 4 36 _________________ / / \/ 4 36 _________________ / \/ 4 36 _________________ / / \/ 4 36 _________________
/ / _______ / / / _______ / / _______ / / / _______ / / _______ / / / _______ / / _______ / / / _______
/ / 1 I*\/ 47919 / / / 1 I*\/ 47919 / / 1 I*\/ 47919 / / / 1 I*\/ 47919 / / 1 I*\/ 47919 / / / 1 I*\/ 47919 / / 1 I*\/ 47919 / / / 1 I*\/ 47919
/ 3*3 / - + ----------- / / 3*3 / - + ----------- / 3*3 / - + ----------- / / 3*3 / - + ----------- / 3*3 / - + ----------- / / 3*3 / - + ----------- / 3*3 / - + ----------- / / 3*3 / - + -----------
\/ \/ 4 36 \/ \/ \/ 4 36 \/ \/ 4 36 \/ \/ \/ 4 36 \/ \/ 4 36 \/ \/ \/ 4 36 \/ \/ 4 36 \/ \/ \/ 4 36
- --------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + - --------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 2 2 2 2
$$\left(\left(\left(- \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} - \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right) + \left(- \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} - \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right)\right) + \left(- \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right)\right) + \left(\frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} - \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right)$$
=
0
$$0$$
producto
/ _______________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________\ / _______________________________________________________________________________________________________________________\ / _________________________________________________________________________________________________________________________\ | / _________________ | | / _________________ | | / _________________ | | / _________________ | | / / _______ | | / / _______ | | / / _______ | | / / _______ | | / / 1 I*\/ 47919 4 20 | | / 4 / 1 I*\/ 47919 20 | | / / 1 I*\/ 47919 4 20 | | / 4 / 1 I*\/ 47919 20 | | I* / 2*3 / - + ----------- + --------------------------------------------------------------- + ------------------------ | | I* / - --------------------------------------------------------------- + 2*3 / - + ----------- + ------------------------ | | I* / 2*3 / - + ----------- + --------------------------------------------------------------- + ------------------------ | | I* / - --------------------------------------------------------------- + 2*3 / - + ----------- + ------------------------ | | _____________________________________________________ / \/ 4 36 _____________________________________________________ _________________ | | _____________________________________________________ / _____________________________________________________ \/ 4 36 _________________ | | _____________________________________________________ / \/ 4 36 _____________________________________________________ _________________ | | _____________________________________________________ / _____________________________________________________ \/ 4 36 _________________ | | / _________________ / / _________________ / _______ | | / _________________ / / _________________ / _______ | | / _________________ / / _________________ / _______ | | / _________________ / / _________________ / _______ | | / / _______ / / / _______ / 1 I*\/ 47919 | | / / _______ / / / _______ / 1 I*\/ 47919 | | / / _______ / / / _______ / 1 I*\/ 47919 | | / / _______ / / / _______ / 1 I*\/ 47919 | | / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- | | / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- | | / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- | | / / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + ----------- | | / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 | | / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 | | / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 | | / 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36 | | / \/ 4 36 _________________ / / \/ 4 36 _________________ | | / \/ 4 36 _________________ / / \/ 4 36 _________________ | | / \/ 4 36 _________________ / / \/ 4 36 _________________ | | / \/ 4 36 _________________ / / \/ 4 36 _________________ | | / / _______ / / / _______ | | / / _______ / / / _______ | | / / _______ / / / _______ | | / / _______ / / / _______ | | / / 1 I*\/ 47919 / / / 1 I*\/ 47919 | | / / 1 I*\/ 47919 / / / 1 I*\/ 47919 | | / / 1 I*\/ 47919 / / / 1 I*\/ 47919 | | / / 1 I*\/ 47919 / / / 1 I*\/ 47919 | | / 3*3 / - + ----------- / / 3*3 / - + ----------- | | / 3*3 / - + ----------- / / 3*3 / - + ----------- | | / 3*3 / - + ----------- / / 3*3 / - + ----------- | | / 3*3 / - + ----------- / / 3*3 / - + ----------- | | \/ \/ 4 36 \/ \/ \/ 4 36 | |\/ \/ 4 36 \/ \/ \/ 4 36 | | \/ \/ 4 36 \/ \/ \/ 4 36 | |\/ \/ 4 36 \/ \/ \/ 4 36 | |- --------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------|*|--------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------|*|- --------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------|*|--------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\left(- \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} - \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right) \left(- \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} - \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right) \left(- \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right) \left(\frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} - \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}\right)$$
=
10
$$10$$
10
Respuesta rápida
[src]
_______________________________________________________________________________________________________________________
/ _________________
/ / _______
/ / 1 I*\/ 47919 4 20
I* / 2*3 / - + ----------- + --------------------------------------------------------------- + ------------------------
_____________________________________________________ / \/ 4 36 _____________________________________________________ _________________
/ _________________ / / _________________ / _______
/ / _______ / / / _______ / 1 I*\/ 47919
/ / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + -----------
/ 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36
/ \/ 4 36 _________________ / / \/ 4 36 _________________
/ / _______ / / / _______
/ / 1 I*\/ 47919 / / / 1 I*\/ 47919
/ 3*3 / - + ----------- / / 3*3 / - + -----------
\/ \/ 4 36 \/ \/ \/ 4 36
x1 = - --------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{1} = - \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} - \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}$$
_________________________________________________________________________________________________________________________
/ _________________
/ / _______
/ 4 / 1 I*\/ 47919 20
I* / - --------------------------------------------------------------- + 2*3 / - + ----------- + ------------------------
_____________________________________________________ / _____________________________________________________ \/ 4 36 _________________
/ _________________ / / _________________ / _______
/ / _______ / / / _______ / 1 I*\/ 47919
/ / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + -----------
/ 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36
/ \/ 4 36 _________________ / / \/ 4 36 _________________
/ / _______ / / / _______
/ / 1 I*\/ 47919 / / / 1 I*\/ 47919
/ 3*3 / - + ----------- / / 3*3 / - + -----------
\/ \/ 4 36 \/ \/ \/ 4 36
x2 = --------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{2} = - \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} - \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}$$
_______________________________________________________________________________________________________________________
/ _________________
/ / _______
/ / 1 I*\/ 47919 4 20
I* / 2*3 / - + ----------- + --------------------------------------------------------------- + ------------------------
_____________________________________________________ / \/ 4 36 _____________________________________________________ _________________
/ _________________ / / _________________ / _______
/ / _______ / / / _______ / 1 I*\/ 47919
/ / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + -----------
/ 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36
/ \/ 4 36 _________________ / / \/ 4 36 _________________
/ / _______ / / / _______
/ / 1 I*\/ 47919 / / / 1 I*\/ 47919
/ 3*3 / - + ----------- / / 3*3 / - + -----------
\/ \/ 4 36 \/ \/ \/ 4 36
x3 = - --------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{3} = - \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}$$
_________________________________________________________________________________________________________________________
/ _________________
/ / _______
/ 4 / 1 I*\/ 47919 20
I* / - --------------------------------------------------------------- + 2*3 / - + ----------- + ------------------------
_____________________________________________________ / _____________________________________________________ \/ 4 36 _________________
/ _________________ / / _________________ / _______
/ / _______ / / / _______ / 1 I*\/ 47919
/ / 1 I*\/ 47919 20 / / / 1 I*\/ 47919 20 3*3 / - + -----------
/ 2*3 / - + ----------- + ------------------------ / / 2*3 / - + ----------- + ------------------------ \/ 4 36
/ \/ 4 36 _________________ / / \/ 4 36 _________________
/ / _______ / / / _______
/ / 1 I*\/ 47919 / / / 1 I*\/ 47919
/ 3*3 / - + ----------- / / 3*3 / - + -----------
\/ \/ 4 36 \/ \/ \/ 4 36
x4 = --------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------
2 2
$$x_{4} = \frac{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2} + \frac{i \sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} - \frac{4}{\sqrt{\frac{20}{3 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}} + 2 \sqrt[3]{\frac{1}{4} + \frac{\sqrt{47919} i}{36}}}}{2}$$
x4 = sqrt(20/(3*(1/4 + sqrt(47919)*i/36)^(1/3)) + 2*(1/4 + sqrt(47919)*i/36)^(1/3))/2 + i*sqrt(20/(3*(1/4 + sqrt(47919)*i/36)^(1/3)) - 4/sqrt(20/(3*(1/4 + sqrt(47919)*i/36)^(1/3)) + 2*(1/4 + sqrt(47919)*i/36)^(1/3)) + 2*(1/4 + sqrt(47919)*i/36)^(1/3))/2
Respuesta numérica
[src]
x1 = 1.26233658131585 + 1.09425900514933*i
x2 = -1.26233658131585 + 1.41052632683902*i
x3 = -1.26233658131585 - 1.41052632683902*i
x4 = 1.26233658131585 - 1.09425900514933*i
x4 = 1.26233658131585 - 1.09425900514933*i