Sr Examen
18*x^3-16*x^2+550*x+144=0 la ecuación
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Solución
Ha introducido
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3 2 18*x - 16*x + 550*x + 144 = 0
$$\left(550 x + \left(18 x^{3} - 16 x^{2}\right)\right) + 144 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(550 x + \left(18 x^{3} - 16 x^{2}\right)\right) + 144 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{8 x^{2}}{9} + \frac{275 x}{9} + 8 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{8}{9}$$
$$q = \frac{c}{a}$$
$$q = \frac{275}{9}$$
$$v = \frac{d}{a}$$
$$v = 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{8}{9}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{275}{9}$$
$$x_{1} x_{2} x_{3} = 8$$
$$\left(550 x + \left(18 x^{3} - 16 x^{2}\right)\right) + 144 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{8 x^{2}}{9} + \frac{275 x}{9} + 8 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{8}{9}$$
$$q = \frac{c}{a}$$
$$q = \frac{275}{9}$$
$$v = \frac{d}{a}$$
$$v = 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{8}{9}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{275}{9}$$
$$x_{1} x_{2} x_{3} = 8$$
Suma y producto de raíces
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suma
________________________ / ________________________ \ ________________________ / ________________________\ ________________________
/ ___________ | / ___________ | / ___________ | / ___________ | / ___________
/ 167320 \/ 585523689 | ___ / 167320 \/ 585523689 | / 167320 \/ 585523689 | ___ / 167320 \/ 585523689 | / 167320 \/ 585523689
3 / ------ + ------------- |\/ 3 *3 / ------ + ------------- ___ | 3 / ------ + ------------- | ___ \/ 3 *3 / ------ + ------------- | 3 / ------ + -------------
8 7361 \/ 729 27 | \/ 729 27 7361*\/ 3 | 8 7361 \/ 729 27 | 7361*\/ 3 \/ 729 27 | 8 \/ 729 27 7361
-- - --------------------------------- + ----------------------------- + I*|----------------------------------- + ---------------------------------| + -- - --------------------------------- + ----------------------------- + I*|- --------------------------------- - -----------------------------------| + -- - ----------------------------- + ---------------------------------
27 ________________________ 6 | 6 ________________________| 27 ________________________ 6 | ________________________ 6 | 27 3 ________________________
/ ___________ | / ___________ | / ___________ | / ___________ | / ___________
/ 167320 \/ 585523689 | / 167320 \/ 585523689 | / 167320 \/ 585523689 | / 167320 \/ 585523689 | / 167320 \/ 585523689
486*3 / ------ + ------------- | 486*3 / ------ + ------------- | 486*3 / ------ + ------------- | 486*3 / ------ + ------------- | 243*3 / ------ + -------------
\/ 729 27 \ \/ 729 27 / \/ 729 27 \ \/ 729 27 / \/ 729 27
$$\left(- \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{3} + \frac{8}{27} + \frac{7361}{243 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right) + \left(\left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right)\right) + \left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right)\right)\right)$$
=
/ ________________________\ / ________________________ \
| / ___________ | | / ___________ |
| ___ / 167320 \/ 585523689 | | ___ / 167320 \/ 585523689 |
| ___ \/ 3 *3 / ------ + ------------- | |\/ 3 *3 / ------ + ------------- ___ |
8 | 7361*\/ 3 \/ 729 27 | | \/ 729 27 7361*\/ 3 |
- + I*|- --------------------------------- - -----------------------------------| + I*|----------------------------------- + ---------------------------------|
9 | ________________________ 6 | | 6 ________________________|
| / ___________ | | / ___________ |
| / 167320 \/ 585523689 | | / 167320 \/ 585523689 |
| 486*3 / ------ + ------------- | | 486*3 / ------ + ------------- |
\ \/ 729 27 / \ \/ 729 27 /
$$\frac{8}{9} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right) + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right)$$
producto
/ ________________________ / ________________________ \\ / ________________________ / ________________________\\ / ________________________ \ | / ___________ | / ___________ || | / ___________ | / ___________ || | / ___________ | | / 167320 \/ 585523689 | ___ / 167320 \/ 585523689 || | / 167320 \/ 585523689 | ___ / 167320 \/ 585523689 || | / 167320 \/ 585523689 | | 3 / ------ + ------------- |\/ 3 *3 / ------ + ------------- ___ || | 3 / ------ + ------------- | ___ \/ 3 *3 / ------ + ------------- || | 3 / ------ + ------------- | |8 7361 \/ 729 27 | \/ 729 27 7361*\/ 3 || |8 7361 \/ 729 27 | 7361*\/ 3 \/ 729 27 || |8 \/ 729 27 7361 | |-- - --------------------------------- + ----------------------------- + I*|----------------------------------- + ---------------------------------||*|-- - --------------------------------- + ----------------------------- + I*|- --------------------------------- - -----------------------------------||*|-- - ----------------------------- + ---------------------------------| |27 ________________________ 6 | 6 ________________________|| |27 ________________________ 6 | ________________________ 6 || |27 3 ________________________| | / ___________ | / ___________ || | / ___________ | / ___________ || | / ___________ | | / 167320 \/ 585523689 | / 167320 \/ 585523689 || | / 167320 \/ 585523689 | / 167320 \/ 585523689 || | / 167320 \/ 585523689 | | 486*3 / ------ + ------------- | 486*3 / ------ + ------------- || | 486*3 / ------ + ------------- | 486*3 / ------ + ------------- || | 243*3 / ------ + ------------- | \ \/ 729 27 \ \/ 729 27 // \ \/ 729 27 \ \/ 729 27 // \ \/ 729 27 /
$$\left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right)\right) \left(- \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{3} + \frac{8}{27} + \frac{7361}{243 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right)$$
=
-8
$$-8$$
-8
Respuesta rápida
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________________________ / ________________________ \
/ ___________ | / ___________ |
/ 167320 \/ 585523689 | ___ / 167320 \/ 585523689 |
3 / ------ + ------------- |\/ 3 *3 / ------ + ------------- ___ |
8 7361 \/ 729 27 | \/ 729 27 7361*\/ 3 |
x1 = -- - --------------------------------- + ----------------------------- + I*|----------------------------------- + ---------------------------------|
27 ________________________ 6 | 6 ________________________|
/ ___________ | / ___________ |
/ 167320 \/ 585523689 | / 167320 \/ 585523689 |
486*3 / ------ + ------------- | 486*3 / ------ + ------------- |
\/ 729 27 \ \/ 729 27 /
$$x_{1} = - \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(\frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6}\right)$$
________________________ / ________________________\
/ ___________ | / ___________ |
/ 167320 \/ 585523689 | ___ / 167320 \/ 585523689 |
3 / ------ + ------------- | ___ \/ 3 *3 / ------ + ------------- |
8 7361 \/ 729 27 | 7361*\/ 3 \/ 729 27 |
x2 = -- - --------------------------------- + ----------------------------- + I*|- --------------------------------- - -----------------------------------|
27 ________________________ 6 | ________________________ 6 |
/ ___________ | / ___________ |
/ 167320 \/ 585523689 | / 167320 \/ 585523689 |
486*3 / ------ + ------------- | 486*3 / ------ + ------------- |
\/ 729 27 \ \/ 729 27 /
$$x_{2} = - \frac{7361}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}} + \frac{8}{27} + \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{6} - \frac{7361 \sqrt{3}}{486 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}\right)$$
________________________
/ ___________
/ 167320 \/ 585523689
3 / ------ + -------------
8 \/ 729 27 7361
x3 = -- - ----------------------------- + ---------------------------------
27 3 ________________________
/ ___________
/ 167320 \/ 585523689
243*3 / ------ + -------------
\/ 729 27
$$x_{3} = - \frac{\sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}{3} + \frac{8}{27} + \frac{7361}{243 \sqrt[3]{\frac{167320}{729} + \frac{\sqrt{585523689}}{27}}}$$
x3 = -(167320/729 + sqrt(585523689)/27)^(1/3)/3 + 8/27 + 7361/(243*(167320/729 + sqrt(585523689)/27)^(1/3))
Respuesta numérica
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x1 = -0.259291807789602
x2 = 0.574090348339245 - 5.52482485478139*i
x3 = 0.574090348339245 + 5.52482485478139*i
x3 = 0.574090348339245 + 5.52482485478139*i