Sr Examen
0.5*x^3+1.36*x^2+5.4x+0.72=0 la ecuación
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Solución
Ha introducido
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3 2 x 34*x 27*x 18 -- + ----- + ---- + -- = 0 2 25 5 25
$$\left(\frac{27 x}{5} + \left(\frac{x^{3}}{2} + \frac{34 x^{2}}{25}\right)\right) + \frac{18}{25} = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(\frac{27 x}{5} + \left(\frac{x^{3}}{2} + \frac{34 x^{2}}{25}\right)\right) + \frac{18}{25} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{68 x^{2}}{25} + \frac{54 x}{5} + \frac{36}{25} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{68}{25}$$
$$q = \frac{c}{a}$$
$$q = \frac{54}{5}$$
$$v = \frac{d}{a}$$
$$v = \frac{36}{25}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{68}{25}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{54}{5}$$
$$x_{1} x_{2} x_{3} = \frac{36}{25}$$
$$\left(\frac{27 x}{5} + \left(\frac{x^{3}}{2} + \frac{34 x^{2}}{25}\right)\right) + \frac{18}{25} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{68 x^{2}}{25} + \frac{54 x}{5} + \frac{36}{25} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{68}{25}$$
$$q = \frac{c}{a}$$
$$q = \frac{54}{5}$$
$$v = \frac{d}{a}$$
$$v = \frac{36}{25}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{68}{25}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{54}{5}$$
$$x_{1} x_{2} x_{3} = \frac{36}{25}$$
Respuesta rápida
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__________________________ / __________________________\
/ __________ | / __________ |
/ 1447318 2*\/ 29185971 | ___ / 1447318 2*\/ 29185971 |
3 / ------- + -------------- | ___ \/ 3 *3 / ------- + -------------- |
68 \/ 421875 1875 7813 | 7813*\/ 3 \/ 421875 1875 |
x1 = - -- - ------------------------------- + ------------------------------------ + I*|- ------------------------------------ - -------------------------------------|
75 2 __________________________ | __________________________ 2 |
/ __________ | / __________ |
/ 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 |
5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- |
\/ 421875 1875 \ \/ 421875 1875 /
$$x_{1} = - \frac{\sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{68}{75} + \frac{7813}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}\right)$$
__________________________ / __________________________ \
/ __________ | / __________ |
/ 1447318 2*\/ 29185971 | ___ / 1447318 2*\/ 29185971 |
3 / ------- + -------------- |\/ 3 *3 / ------- + -------------- ___ |
68 \/ 421875 1875 7813 | \/ 421875 1875 7813*\/ 3 |
x2 = - -- - ------------------------------- + ------------------------------------ + I*|------------------------------------- + ------------------------------------|
75 2 __________________________ | 2 __________________________|
/ __________ | / __________ |
/ 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 |
5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- |
\/ 421875 1875 \ \/ 421875 1875 /
$$x_{2} = - \frac{\sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{68}{75} + \frac{7813}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + i \left(\frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2}\right)$$
__________________________
/ __________
68 / 1447318 2*\/ 29185971 15626
x3 = - -- + 3 / ------- + -------------- - ------------------------------------
75 \/ 421875 1875 __________________________
/ __________
/ 1447318 2*\/ 29185971
5625*3 / ------- + --------------
\/ 421875 1875
$$x_{3} = - \frac{15626}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} - \frac{68}{75} + \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}$$
x3 = -15626/(5625*(1447318/421875 + 2*sqrt(29185971)/1875)^(1/3)) - 68/75 + (1447318/421875 + 2*sqrt(29185971)/1875)^(1/3)
Suma y producto de raíces
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suma
__________________________ / __________________________\ __________________________ / __________________________ \
/ __________ | / __________ | / __________ | / __________ |
/ 1447318 2*\/ 29185971 | ___ / 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 | ___ / 1447318 2*\/ 29185971 | __________________________
3 / ------- + -------------- | ___ \/ 3 *3 / ------- + -------------- | 3 / ------- + -------------- |\/ 3 *3 / ------- + -------------- ___ | / __________
68 \/ 421875 1875 7813 | 7813*\/ 3 \/ 421875 1875 | 68 \/ 421875 1875 7813 | \/ 421875 1875 7813*\/ 3 | 68 / 1447318 2*\/ 29185971 15626
- -- - ------------------------------- + ------------------------------------ + I*|- ------------------------------------ - -------------------------------------| + - -- - ------------------------------- + ------------------------------------ + I*|------------------------------------- + ------------------------------------| + - -- + 3 / ------- + -------------- - ------------------------------------
75 2 __________________________ | __________________________ 2 | 75 2 __________________________ | 2 __________________________| 75 \/ 421875 1875 __________________________
/ __________ | / __________ | / __________ | / __________ | / __________
/ 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971
5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- | 5625*3 / ------- + --------------
\/ 421875 1875 \ \/ 421875 1875 / \/ 421875 1875 \ \/ 421875 1875 / \/ 421875 1875
$$\left(- \frac{15626}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} - \frac{68}{75} + \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{68}{75} + \frac{7813}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{68}{75} + \frac{7813}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + i \left(\frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2}\right)\right)\right)$$
=
/ __________________________ \ / __________________________\
| / __________ | | / __________ |
| ___ / 1447318 2*\/ 29185971 | | ___ / 1447318 2*\/ 29185971 |
|\/ 3 *3 / ------- + -------------- ___ | | ___ \/ 3 *3 / ------- + -------------- |
68 | \/ 421875 1875 7813*\/ 3 | | 7813*\/ 3 \/ 421875 1875 |
- -- + I*|------------------------------------- + ------------------------------------| + I*|- ------------------------------------ - -------------------------------------|
25 | 2 __________________________| | __________________________ 2 |
| / __________ | | / __________ |
| / 1447318 2*\/ 29185971 | | / 1447318 2*\/ 29185971 |
| 5625*3 / ------- + -------------- | | 5625*3 / ------- + -------------- |
\ \/ 421875 1875 / \ \/ 421875 1875 /
$$- \frac{68}{25} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}\right) + i \left(\frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2}\right)$$
producto
/ __________________________ / __________________________\\ / __________________________ / __________________________ \\ | / __________ | / __________ || | / __________ | / __________ || | / 1447318 2*\/ 29185971 | ___ / 1447318 2*\/ 29185971 || | / 1447318 2*\/ 29185971 | ___ / 1447318 2*\/ 29185971 || / __________________________ \ | 3 / ------- + -------------- | ___ \/ 3 *3 / ------- + -------------- || | 3 / ------- + -------------- |\/ 3 *3 / ------- + -------------- ___ || | / __________ | | 68 \/ 421875 1875 7813 | 7813*\/ 3 \/ 421875 1875 || | 68 \/ 421875 1875 7813 | \/ 421875 1875 7813*\/ 3 || | 68 / 1447318 2*\/ 29185971 15626 | |- -- - ------------------------------- + ------------------------------------ + I*|- ------------------------------------ - -------------------------------------||*|- -- - ------------------------------- + ------------------------------------ + I*|------------------------------------- + ------------------------------------||*|- -- + 3 / ------- + -------------- - ------------------------------------| | 75 2 __________________________ | __________________________ 2 || | 75 2 __________________________ | 2 __________________________|| | 75 \/ 421875 1875 __________________________| | / __________ | / __________ || | / __________ | / __________ || | / __________ | | / 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 || | / 1447318 2*\/ 29185971 | / 1447318 2*\/ 29185971 || | / 1447318 2*\/ 29185971 | | 5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- || | 5625*3 / ------- + -------------- | 5625*3 / ------- + -------------- || | 5625*3 / ------- + -------------- | \ \/ 421875 1875 \ \/ 421875 1875 // \ \/ 421875 1875 \ \/ 421875 1875 // \ \/ 421875 1875 /
$$\left(- \frac{\sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{68}{75} + \frac{7813}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + i \left(\frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{68}{75} + \frac{7813}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}{2} - \frac{7813 \sqrt{3}}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}}\right)\right) \left(- \frac{15626}{5625 \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}} - \frac{68}{75} + \sqrt[3]{\frac{1447318}{421875} + \frac{2 \sqrt{29185971}}{1875}}\right)$$
=
-36 ---- 25
$$- \frac{36}{25}$$
-36/25
Respuesta numérica
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x1 = -1.29106077121795 - 2.96262436861442*i
x2 = -1.29106077121795 + 2.96262436861442*i
x3 = -0.137878457564102
x3 = -0.137878457564102