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Expresiones idénticas
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Expresiones semejantes
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x^3-14x^2+14x-15=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - 14*x + 14*x - 15 = 0
$$\left(14 x + \left(x^{3} - 14 x^{2}\right)\right) - 15 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -14$$
$$q = \frac{c}{a}$$
$$q = 14$$
$$v = \frac{d}{a}$$
$$v = -15$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 14$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 14$$
$$x_{1} x_{2} x_{3} = -15$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -14$$
$$q = \frac{c}{a}$$
$$q = 14$$
$$v = \frac{d}{a}$$
$$v = -15$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 14$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 14$$
$$x_{1} x_{2} x_{3} = -15$$
Respuesta rápida
[src]
___________________ / ___________________ \
/ ________ | / ________ |
/ 4129 \/ 271065 | ___ / 4129 \/ 271065 |
3 / ---- + ---------- | \/ 3 *3 / ---- + ---------- ___ |
14 77 \/ 54 18 | \/ 54 18 77*\/ 3 |
x1 = -- - -------------------------- - ------------------------ + I*|- ------------------------------ + --------------------------|
3 ___________________ 2 | 2 ___________________|
/ ________ | / ________ |
/ 4129 \/ 271065 | / 4129 \/ 271065 |
9*3 / ---- + ---------- | 9*3 / ---- + ---------- |
\/ 54 18 \ \/ 54 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} - \frac{77}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} + \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}\right)$$
___________________ / ___________________ \
/ ________ | / ________ |
/ 4129 \/ 271065 | ___ / 4129 \/ 271065 |
3 / ---- + ---------- |\/ 3 *3 / ---- + ---------- ___ |
14 77 \/ 54 18 | \/ 54 18 77*\/ 3 |
x2 = -- - -------------------------- - ------------------------ + I*|------------------------------ - --------------------------|
3 ___________________ 2 | 2 ___________________|
/ ________ | / ________ |
/ 4129 \/ 271065 | / 4129 \/ 271065 |
9*3 / ---- + ---------- | 9*3 / ---- + ---------- |
\/ 54 18 \ \/ 54 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} - \frac{77}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + i \left(- \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2}\right)$$
___________________
/ ________
14 / 4129 \/ 271065 154
x3 = -- + 3 / ---- + ---------- + --------------------------
3 \/ 54 18 ___________________
/ ________
/ 4129 \/ 271065
9*3 / ---- + ----------
\/ 54 18
$$x_{3} = \frac{154}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}$$
x3 = 154/(9*(sqrt(271065)/18 + 4129/54)^(1/3)) + 14/3 + (sqrt(271065)/18 + 4129/54)^(1/3)
Suma y producto de raíces
[src]
suma
___________________ / ___________________ \ ___________________ / ___________________ \
/ ________ | / ________ | / ________ | / ________ |
/ 4129 \/ 271065 | ___ / 4129 \/ 271065 | / 4129 \/ 271065 | ___ / 4129 \/ 271065 | ___________________
3 / ---- + ---------- | \/ 3 *3 / ---- + ---------- ___ | 3 / ---- + ---------- |\/ 3 *3 / ---- + ---------- ___ | / ________
14 77 \/ 54 18 | \/ 54 18 77*\/ 3 | 14 77 \/ 54 18 | \/ 54 18 77*\/ 3 | 14 / 4129 \/ 271065 154
-- - -------------------------- - ------------------------ + I*|- ------------------------------ + --------------------------| + -- - -------------------------- - ------------------------ + I*|------------------------------ - --------------------------| + -- + 3 / ---- + ---------- + --------------------------
3 ___________________ 2 | 2 ___________________| 3 ___________________ 2 | 2 ___________________| 3 \/ 54 18 ___________________
/ ________ | / ________ | / ________ | / ________ | / ________
/ 4129 \/ 271065 | / 4129 \/ 271065 | / 4129 \/ 271065 | / 4129 \/ 271065 | / 4129 \/ 271065
9*3 / ---- + ---------- | 9*3 / ---- + ---------- | 9*3 / ---- + ---------- | 9*3 / ---- + ---------- | 9*3 / ---- + ----------
\/ 54 18 \ \/ 54 18 / \/ 54 18 \ \/ 54 18 / \/ 54 18
$$\left(\frac{154}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} - \frac{77}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} + \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} - \frac{77}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + i \left(- \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2}\right)\right)\right)$$
=
/ ___________________ \ / ___________________ \
| / ________ | | / ________ |
| ___ / 4129 \/ 271065 | | ___ / 4129 \/ 271065 |
|\/ 3 *3 / ---- + ---------- ___ | | \/ 3 *3 / ---- + ---------- ___ |
| \/ 54 18 77*\/ 3 | | \/ 54 18 77*\/ 3 |
14 + I*|------------------------------ - --------------------------| + I*|- ------------------------------ + --------------------------|
| 2 ___________________| | 2 ___________________|
| / ________ | | / ________ |
| / 4129 \/ 271065 | | / 4129 \/ 271065 |
| 9*3 / ---- + ---------- | | 9*3 / ---- + ---------- |
\ \/ 54 18 / \ \/ 54 18 /
$$14 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} + \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}\right) + i \left(- \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2}\right)$$
producto
/ ___________________ / ___________________ \\ / ___________________ / ___________________ \\ | / ________ | / ________ || | / ________ | / ________ || | / 4129 \/ 271065 | ___ / 4129 \/ 271065 || | / 4129 \/ 271065 | ___ / 4129 \/ 271065 || / ___________________ \ | 3 / ---- + ---------- | \/ 3 *3 / ---- + ---------- ___ || | 3 / ---- + ---------- |\/ 3 *3 / ---- + ---------- ___ || | / ________ | |14 77 \/ 54 18 | \/ 54 18 77*\/ 3 || |14 77 \/ 54 18 | \/ 54 18 77*\/ 3 || |14 / 4129 \/ 271065 154 | |-- - -------------------------- - ------------------------ + I*|- ------------------------------ + --------------------------||*|-- - -------------------------- - ------------------------ + I*|------------------------------ - --------------------------||*|-- + 3 / ---- + ---------- + --------------------------| |3 ___________________ 2 | 2 ___________________|| |3 ___________________ 2 | 2 ___________________|| |3 \/ 54 18 ___________________| | / ________ | / ________ || | / ________ | / ________ || | / ________ | | / 4129 \/ 271065 | / 4129 \/ 271065 || | / 4129 \/ 271065 | / 4129 \/ 271065 || | / 4129 \/ 271065 | | 9*3 / ---- + ---------- | 9*3 / ---- + ---------- || | 9*3 / ---- + ---------- | 9*3 / ---- + ---------- || | 9*3 / ---- + ---------- | \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} - \frac{77}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + i \left(- \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} - \frac{77}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}{2} + \frac{77 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}}\right)\right) \left(\frac{154}{9 \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}} + \frac{14}{3} + \sqrt[3]{\frac{\sqrt{271065}}{18} + \frac{4129}{54}}\right)$$
=
15
$$15$$
15
Respuesta numérica
[src]
x1 = 13.012713104266
x2 = 0.493643447866982 + 0.953433280618269*i
x3 = 0.493643447866982 - 0.953433280618269*i
x3 = 0.493643447866982 - 0.953433280618269*i