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Expresiones semejantes
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x^3-4*x^2-25*x-100=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - 4*x - 25*x - 100 = 0
$$\left(- 25 x + \left(x^{3} - 4 x^{2}\right)\right) - 100 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -25$$
$$v = \frac{d}{a}$$
$$v = -100$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -25$$
$$x_{1} x_{2} x_{3} = -100$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -25$$
$$v = \frac{d}{a}$$
$$v = -100$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -25$$
$$x_{1} x_{2} x_{3} = -100$$
Respuesta rápida
[src]
____________________ / ____________________ \
/ _______ | / _______ |
/ 1864 5*\/ 12093 | ___ / 1864 5*\/ 12093 |
3 / ---- + ----------- | \/ 3 *3 / ---- + ----------- ___ |
4 91 \/ 27 9 | \/ 27 9 91*\/ 3 |
x1 = - - ---------------------------- - ------------------------- + I*|- ------------------------------- + ----------------------------|
3 ____________________ 2 | 2 ____________________|
/ _______ | / _______ |
/ 1864 5*\/ 12093 | / 1864 5*\/ 12093 |
18*3 / ---- + ----------- | 18*3 / ---- + ----------- |
\/ 27 9 \ \/ 27 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} - \frac{91}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} + \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}\right)$$
____________________ / ____________________ \
/ _______ | / _______ |
/ 1864 5*\/ 12093 | ___ / 1864 5*\/ 12093 |
3 / ---- + ----------- |\/ 3 *3 / ---- + ----------- ___ |
4 91 \/ 27 9 | \/ 27 9 91*\/ 3 |
x2 = - - ---------------------------- - ------------------------- + I*|------------------------------- - ----------------------------|
3 ____________________ 2 | 2 ____________________|
/ _______ | / _______ |
/ 1864 5*\/ 12093 | / 1864 5*\/ 12093 |
18*3 / ---- + ----------- | 18*3 / ---- + ----------- |
\/ 27 9 \ \/ 27 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} - \frac{91}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{4}{3} + i \left(- \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2}\right)$$
____________________
/ _______
4 / 1864 5*\/ 12093 91
x3 = - + 3 / ---- + ----------- + ---------------------------
3 \/ 27 9 ____________________
/ _______
/ 1864 5*\/ 12093
9*3 / ---- + -----------
\/ 27 9
$$x_{3} = \frac{4}{3} + \frac{91}{9 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}$$
x3 = 4/3 + 91/(9*(5*sqrt(12093)/9 + 1864/27)^(1/3)) + (5*sqrt(12093)/9 + 1864/27)^(1/3)
Suma y producto de raíces
[src]
suma
____________________ / ____________________ \ ____________________ / ____________________ \
/ _______ | / _______ | / _______ | / _______ |
/ 1864 5*\/ 12093 | ___ / 1864 5*\/ 12093 | / 1864 5*\/ 12093 | ___ / 1864 5*\/ 12093 | ____________________
3 / ---- + ----------- | \/ 3 *3 / ---- + ----------- ___ | 3 / ---- + ----------- |\/ 3 *3 / ---- + ----------- ___ | / _______
4 91 \/ 27 9 | \/ 27 9 91*\/ 3 | 4 91 \/ 27 9 | \/ 27 9 91*\/ 3 | 4 / 1864 5*\/ 12093 91
- - ---------------------------- - ------------------------- + I*|- ------------------------------- + ----------------------------| + - - ---------------------------- - ------------------------- + I*|------------------------------- - ----------------------------| + - + 3 / ---- + ----------- + ---------------------------
3 ____________________ 2 | 2 ____________________| 3 ____________________ 2 | 2 ____________________| 3 \/ 27 9 ____________________
/ _______ | / _______ | / _______ | / _______ | / _______
/ 1864 5*\/ 12093 | / 1864 5*\/ 12093 | / 1864 5*\/ 12093 | / 1864 5*\/ 12093 | / 1864 5*\/ 12093
18*3 / ---- + ----------- | 18*3 / ---- + ----------- | 18*3 / ---- + ----------- | 18*3 / ---- + ----------- | 9*3 / ---- + -----------
\/ 27 9 \ \/ 27 9 / \/ 27 9 \ \/ 27 9 / \/ 27 9
$$\left(\frac{4}{3} + \frac{91}{9 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} - \frac{91}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} + \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} - \frac{91}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{4}{3} + i \left(- \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2}\right)\right)\right)$$
=
/ ____________________ \ / ____________________ \
| / _______ | | / _______ |
| ___ / 1864 5*\/ 12093 | | ___ / 1864 5*\/ 12093 |
|\/ 3 *3 / ---- + ----------- ___ | | \/ 3 *3 / ---- + ----------- ___ |
| \/ 27 9 91*\/ 3 | | \/ 27 9 91*\/ 3 |
4 + I*|------------------------------- - ----------------------------| + I*|- ------------------------------- + ----------------------------|
| 2 ____________________| | 2 ____________________|
| / _______ | | / _______ |
| / 1864 5*\/ 12093 | | / 1864 5*\/ 12093 |
| 18*3 / ---- + ----------- | | 18*3 / ---- + ----------- |
\ \/ 27 9 / \ \/ 27 9 /
$$4 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} + \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}\right) + i \left(- \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2}\right)$$
producto
/ ____________________ / ____________________ \\ / ____________________ / ____________________ \\ | / _______ | / _______ || | / _______ | / _______ || | / 1864 5*\/ 12093 | ___ / 1864 5*\/ 12093 || | / 1864 5*\/ 12093 | ___ / 1864 5*\/ 12093 || / ____________________ \ | 3 / ---- + ----------- | \/ 3 *3 / ---- + ----------- ___ || | 3 / ---- + ----------- |\/ 3 *3 / ---- + ----------- ___ || | / _______ | |4 91 \/ 27 9 | \/ 27 9 91*\/ 3 || |4 91 \/ 27 9 | \/ 27 9 91*\/ 3 || |4 / 1864 5*\/ 12093 91 | |- - ---------------------------- - ------------------------- + I*|- ------------------------------- + ----------------------------||*|- - ---------------------------- - ------------------------- + I*|------------------------------- - ----------------------------||*|- + 3 / ---- + ----------- + ---------------------------| |3 ____________________ 2 | 2 ____________________|| |3 ____________________ 2 | 2 ____________________|| |3 \/ 27 9 ____________________| | / _______ | / _______ || | / _______ | / _______ || | / _______ | | / 1864 5*\/ 12093 | / 1864 5*\/ 12093 || | / 1864 5*\/ 12093 | / 1864 5*\/ 12093 || | / 1864 5*\/ 12093 | | 18*3 / ---- + ----------- | 18*3 / ---- + ----------- || | 18*3 / ---- + ----------- | 18*3 / ---- + ----------- || | 9*3 / ---- + ----------- | \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 /
$$\left(- \frac{\sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} - \frac{91}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{4}{3} + i \left(- \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} - \frac{91}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}{2} + \frac{91 \sqrt{3}}{18 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}}\right)\right) \left(\frac{4}{3} + \frac{91}{9 \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}} + \sqrt[3]{\frac{5 \sqrt{12093}}{9} + \frac{1864}{27}}\right)$$
=
100
$$100$$
100
Respuesta numérica
[src]
x1 = -2.19805691282578 - 2.66060479565639*i
x2 = -2.19805691282578 + 2.66060479565639*i
x3 = 8.39611382565155
x3 = 8.39611382565155