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Expresiones idénticas
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Expresiones semejantes
- -x^3+5x^2-8,23x-6.669=0
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- x^3+5x^2-8,23x+6.669=0
-x^3+5x^2-8,23x+6.669=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 823*x 6669
- x + 5*x - ----- + ---- = 0
100 1000
$$\left(- \frac{823 x}{100} + \left(- x^{3} + 5 x^{2}\right)\right) + \frac{6669}{1000} = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- \frac{823 x}{100} + \left(- x^{3} + 5 x^{2}\right)\right) + \frac{6669}{1000} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 5 x^{2} + \frac{823 x}{100} - \frac{6669}{1000} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = \frac{823}{100}$$
$$v = \frac{d}{a}$$
$$v = - \frac{6669}{1000}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{823}{100}$$
$$x_{1} x_{2} x_{3} = - \frac{6669}{1000}$$
$$\left(- \frac{823 x}{100} + \left(- x^{3} + 5 x^{2}\right)\right) + \frac{6669}{1000} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 5 x^{2} + \frac{823 x}{100} - \frac{6669}{1000} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = \frac{823}{100}$$
$$v = \frac{d}{a}$$
$$v = - \frac{6669}{1000}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{823}{100}$$
$$x_{1} x_{2} x_{3} = - \frac{6669}{1000}$$
Respuesta rápida
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______________________ / ______________________ \
/ __________ | / __________ |
/ 59713 \/ 44018805 | ___ / 59713 \/ 44018805 |
3 / ----- + ------------ | \/ 3 *3 / ----- + ------------ ___ |
5 31 \/ 54000 6000 | \/ 54000 6000 31*\/ 3 |
x1 = - - -------------------------------- - --------------------------- + I*|- --------------------------------- + --------------------------------|
3 ______________________ 2 | 2 ______________________|
/ __________ | / __________ |
/ 59713 \/ 44018805 | / 59713 \/ 44018805 |
1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ |
\/ 54000 6000 \ \/ 54000 6000 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} - \frac{31}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} + \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}\right)$$
______________________ / ______________________ \
/ __________ | / __________ |
/ 59713 \/ 44018805 | ___ / 59713 \/ 44018805 |
3 / ----- + ------------ |\/ 3 *3 / ----- + ------------ ___ |
5 31 \/ 54000 6000 | \/ 54000 6000 31*\/ 3 |
x2 = - - -------------------------------- - --------------------------- + I*|--------------------------------- - --------------------------------|
3 ______________________ 2 | 2 ______________________|
/ __________ | / __________ |
/ 59713 \/ 44018805 | / 59713 \/ 44018805 |
1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ |
\/ 54000 6000 \ \/ 54000 6000 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} - \frac{31}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{5}{3} + i \left(- \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2}\right)$$
______________________
/ __________
5 / 59713 \/ 44018805 31
x3 = - + 3 / ----- + ------------ + -------------------------------
3 \/ 54000 6000 ______________________
/ __________
/ 59713 \/ 44018805
900*3 / ----- + ------------
\/ 54000 6000
$$x_{3} = \frac{31}{900 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}} + \frac{5}{3}$$
x3 = 31/(900*(sqrt(44018805)/6000 + 59713/54000)^(1/3)) + (sqrt(44018805)/6000 + 59713/54000)^(1/3) + 5/3
Suma y producto de raíces
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suma
______________________ / ______________________ \ ______________________ / ______________________ \
/ __________ | / __________ | / __________ | / __________ |
/ 59713 \/ 44018805 | ___ / 59713 \/ 44018805 | / 59713 \/ 44018805 | ___ / 59713 \/ 44018805 | ______________________
3 / ----- + ------------ | \/ 3 *3 / ----- + ------------ ___ | 3 / ----- + ------------ |\/ 3 *3 / ----- + ------------ ___ | / __________
5 31 \/ 54000 6000 | \/ 54000 6000 31*\/ 3 | 5 31 \/ 54000 6000 | \/ 54000 6000 31*\/ 3 | 5 / 59713 \/ 44018805 31
- - -------------------------------- - --------------------------- + I*|- --------------------------------- + --------------------------------| + - - -------------------------------- - --------------------------- + I*|--------------------------------- - --------------------------------| + - + 3 / ----- + ------------ + -------------------------------
3 ______________________ 2 | 2 ______________________| 3 ______________________ 2 | 2 ______________________| 3 \/ 54000 6000 ______________________
/ __________ | / __________ | / __________ | / __________ | / __________
/ 59713 \/ 44018805 | / 59713 \/ 44018805 | / 59713 \/ 44018805 | / 59713 \/ 44018805 | / 59713 \/ 44018805
1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ | 900*3 / ----- + ------------
\/ 54000 6000 \ \/ 54000 6000 / \/ 54000 6000 \ \/ 54000 6000 / \/ 54000 6000
$$\left(\frac{31}{900 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}} + \frac{5}{3}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} - \frac{31}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} + \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} - \frac{31}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{5}{3} + i \left(- \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2}\right)\right)\right)$$
=
/ ______________________ \ / ______________________ \
| / __________ | | / __________ |
| ___ / 59713 \/ 44018805 | | ___ / 59713 \/ 44018805 |
|\/ 3 *3 / ----- + ------------ ___ | | \/ 3 *3 / ----- + ------------ ___ |
| \/ 54000 6000 31*\/ 3 | | \/ 54000 6000 31*\/ 3 |
5 + I*|--------------------------------- - --------------------------------| + I*|- --------------------------------- + --------------------------------|
| 2 ______________________| | 2 ______________________|
| / __________ | | / __________ |
| / 59713 \/ 44018805 | | / 59713 \/ 44018805 |
| 1800*3 / ----- + ------------ | | 1800*3 / ----- + ------------ |
\ \/ 54000 6000 / \ \/ 54000 6000 /
$$5 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} + \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}\right) + i \left(- \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2}\right)$$
producto
/ ______________________ / ______________________ \\ / ______________________ / ______________________ \\ | / __________ | / __________ || | / __________ | / __________ || | / 59713 \/ 44018805 | ___ / 59713 \/ 44018805 || | / 59713 \/ 44018805 | ___ / 59713 \/ 44018805 || / ______________________ \ | 3 / ----- + ------------ | \/ 3 *3 / ----- + ------------ ___ || | 3 / ----- + ------------ |\/ 3 *3 / ----- + ------------ ___ || | / __________ | |5 31 \/ 54000 6000 | \/ 54000 6000 31*\/ 3 || |5 31 \/ 54000 6000 | \/ 54000 6000 31*\/ 3 || |5 / 59713 \/ 44018805 31 | |- - -------------------------------- - --------------------------- + I*|- --------------------------------- + --------------------------------||*|- - -------------------------------- - --------------------------- + I*|--------------------------------- - --------------------------------||*|- + 3 / ----- + ------------ + -------------------------------| |3 ______________________ 2 | 2 ______________________|| |3 ______________________ 2 | 2 ______________________|| |3 \/ 54000 6000 ______________________| | / __________ | / __________ || | / __________ | / __________ || | / __________ | | / 59713 \/ 44018805 | / 59713 \/ 44018805 || | / 59713 \/ 44018805 | / 59713 \/ 44018805 || | / 59713 \/ 44018805 | | 1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ || | 1800*3 / ----- + ------------ | 1800*3 / ----- + ------------ || | 900*3 / ----- + ------------ | \ \/ 54000 6000 \ \/ 54000 6000 // \ \/ 54000 6000 \ \/ 54000 6000 // \ \/ 54000 6000 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} - \frac{31}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{5}{3} + i \left(- \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} - \frac{31}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}{2} + \frac{31 \sqrt{3}}{1800 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}}\right)\right) \left(\frac{31}{900 \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}}} + \sqrt[3]{\frac{\sqrt{44018805}}{6000} + \frac{59713}{54000}} + \frac{5}{3}\right)$$
=
6669 ---- 1000
$$\frac{6669}{1000}$$
6669/1000
Respuesta numérica
[src]
x1 = 2.99597230901717
x2 = 1.00201384549142 - 1.10542154165518*i
x3 = 1.00201384549142 + 1.10542154165518*i
x3 = 1.00201384549142 + 1.10542154165518*i