Sr Examen
2*x^2-13*x^3+20x^2+14x-37=0 la ecuación
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Solución
Ha introducido
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2 3 2 2*x - 13*x + 20*x + 14*x - 37 = 0
$$\left(14 x + \left(20 x^{2} + \left(- 13 x^{3} + 2 x^{2}\right)\right)\right) - 37 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(14 x + \left(20 x^{2} + \left(- 13 x^{3} + 2 x^{2}\right)\right)\right) - 37 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{22 x^{2}}{13} - \frac{14 x}{13} + \frac{37}{13} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{22}{13}$$
$$q = \frac{c}{a}$$
$$q = - \frac{14}{13}$$
$$v = \frac{d}{a}$$
$$v = \frac{37}{13}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{22}{13}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{14}{13}$$
$$x_{1} x_{2} x_{3} = \frac{37}{13}$$
$$\left(14 x + \left(20 x^{2} + \left(- 13 x^{3} + 2 x^{2}\right)\right)\right) - 37 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{22 x^{2}}{13} - \frac{14 x}{13} + \frac{37}{13} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{22}{13}$$
$$q = \frac{c}{a}$$
$$q = - \frac{14}{13}$$
$$v = \frac{d}{a}$$
$$v = \frac{37}{13}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{22}{13}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{14}{13}$$
$$x_{1} x_{2} x_{3} = \frac{37}{13}$$
Respuesta rápida
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________________________ / ________________________\
/ _________ | / _________ |
/ 111499 3*\/ 5299881 | ___ / 111499 3*\/ 5299881 |
3 / ------ + ------------- | ___ \/ 3 *3 / ------ + ------------- |
22 \/ 4394 338 515 | 515*\/ 3 \/ 4394 338 |
x1 = -- + ----------------------------- + --------------------------------- + I*|- --------------------------------- + -----------------------------------|
39 6 ________________________ | ________________________ 6 |
/ _________ | / _________ |
/ 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 |
507*3 / ------ + ------------- | 507*3 / ------ + ------------- |
\/ 4394 338 \ \/ 4394 338 /
$$x_{1} = \frac{515}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39} + \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + i \left(- \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6}\right)$$
________________________ / ________________________ \
/ _________ | / _________ |
/ 111499 3*\/ 5299881 | ___ / 111499 3*\/ 5299881 |
3 / ------ + ------------- | \/ 3 *3 / ------ + ------------- ___ |
22 \/ 4394 338 515 | \/ 4394 338 515*\/ 3 |
x2 = -- + ----------------------------- + --------------------------------- + I*|- ----------------------------------- + ---------------------------------|
39 6 ________________________ | 6 ________________________|
/ _________ | / _________ |
/ 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 |
507*3 / ------ + ------------- | 507*3 / ------ + ------------- |
\/ 4394 338 \ \/ 4394 338 /
$$x_{2} = \frac{515}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39} + \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}\right)$$
________________________
/ _________
/ 111499 3*\/ 5299881
3 / ------ + -------------
22 1030 \/ 4394 338
x3 = -- - --------------------------------- - -----------------------------
39 ________________________ 3
/ _________
/ 111499 3*\/ 5299881
507*3 / ------ + -------------
\/ 4394 338
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{3} - \frac{1030}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39}$$
x3 = -(3*sqrt(5299881)/338 + 111499/4394)^(1/3)/3 - 1030/(507*(3*sqrt(5299881)/338 + 111499/4394)^(1/3)) + 22/39
Suma y producto de raíces
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suma
________________________ / ________________________\ ________________________ / ________________________ \ ________________________
/ _________ | / _________ | / _________ | / _________ | / _________
/ 111499 3*\/ 5299881 | ___ / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 | ___ / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881
3 / ------ + ------------- | ___ \/ 3 *3 / ------ + ------------- | 3 / ------ + ------------- | \/ 3 *3 / ------ + ------------- ___ | 3 / ------ + -------------
22 \/ 4394 338 515 | 515*\/ 3 \/ 4394 338 | 22 \/ 4394 338 515 | \/ 4394 338 515*\/ 3 | 22 1030 \/ 4394 338
-- + ----------------------------- + --------------------------------- + I*|- --------------------------------- + -----------------------------------| + -- + ----------------------------- + --------------------------------- + I*|- ----------------------------------- + ---------------------------------| + -- - --------------------------------- - -----------------------------
39 6 ________________________ | ________________________ 6 | 39 6 ________________________ | 6 ________________________| 39 ________________________ 3
/ _________ | / _________ | / _________ | / _________ | / _________
/ 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881
507*3 / ------ + ------------- | 507*3 / ------ + ------------- | 507*3 / ------ + ------------- | 507*3 / ------ + ------------- | 507*3 / ------ + -------------
\/ 4394 338 \ \/ 4394 338 / \/ 4394 338 \ \/ 4394 338 / \/ 4394 338
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{3} - \frac{1030}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39}\right) + \left(\left(\frac{515}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39} + \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}\right)\right) + \left(\frac{515}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39} + \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + i \left(- \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6}\right)\right)\right)$$
=
/ ________________________\ / ________________________ \
| / _________ | | / _________ |
| ___ / 111499 3*\/ 5299881 | | ___ / 111499 3*\/ 5299881 |
| ___ \/ 3 *3 / ------ + ------------- | | \/ 3 *3 / ------ + ------------- ___ |
22 | 515*\/ 3 \/ 4394 338 | | \/ 4394 338 515*\/ 3 |
-- + I*|- --------------------------------- + -----------------------------------| + I*|- ----------------------------------- + ---------------------------------|
13 | ________________________ 6 | | 6 ________________________|
| / _________ | | / _________ |
| / 111499 3*\/ 5299881 | | / 111499 3*\/ 5299881 |
| 507*3 / ------ + ------------- | | 507*3 / ------ + ------------- |
\ \/ 4394 338 / \ \/ 4394 338 /
$$\frac{22}{13} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}\right) + i \left(- \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6}\right)$$
producto
/ ________________________ / ________________________\\ / ________________________ / ________________________ \\ / ________________________\ | / _________ | / _________ || | / _________ | / _________ || | / _________ | | / 111499 3*\/ 5299881 | ___ / 111499 3*\/ 5299881 || | / 111499 3*\/ 5299881 | ___ / 111499 3*\/ 5299881 || | / 111499 3*\/ 5299881 | | 3 / ------ + ------------- | ___ \/ 3 *3 / ------ + ------------- || | 3 / ------ + ------------- | \/ 3 *3 / ------ + ------------- ___ || | 3 / ------ + ------------- | |22 \/ 4394 338 515 | 515*\/ 3 \/ 4394 338 || |22 \/ 4394 338 515 | \/ 4394 338 515*\/ 3 || |22 1030 \/ 4394 338 | |-- + ----------------------------- + --------------------------------- + I*|- --------------------------------- + -----------------------------------||*|-- + ----------------------------- + --------------------------------- + I*|- ----------------------------------- + ---------------------------------||*|-- - --------------------------------- - -----------------------------| |39 6 ________________________ | ________________________ 6 || |39 6 ________________________ | 6 ________________________|| |39 ________________________ 3 | | / _________ | / _________ || | / _________ | / _________ || | / _________ | | / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 || | / 111499 3*\/ 5299881 | / 111499 3*\/ 5299881 || | / 111499 3*\/ 5299881 | | 507*3 / ------ + ------------- | 507*3 / ------ + ------------- || | 507*3 / ------ + ------------- | 507*3 / ------ + ------------- || | 507*3 / ------ + ------------- | \ \/ 4394 338 \ \/ 4394 338 // \ \/ 4394 338 \ \/ 4394 338 // \ \/ 4394 338 /
$$\left(\frac{515}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39} + \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + i \left(- \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6}\right)\right) \left(\frac{515}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39} + \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{6} + \frac{515 \sqrt{3}}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}}{3} - \frac{1030}{507 \sqrt[3]{\frac{3 \sqrt{5299881}}{338} + \frac{111499}{4394}}} + \frac{22}{39}\right)$$
=
-37 ---- 13
$$- \frac{37}{13}$$
-37/13
Respuesta numérica
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x1 = 1.44433778143318 + 0.541187516562304*i
x2 = 1.44433778143318 - 0.541187516562304*i
x3 = -1.19636787055866
x3 = -1.19636787055866