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Expresiones idénticas
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Expresiones semejantes
- -x^3+4*x^2+15*x+30=0
- -x^3-4*x^2-15*x+30=0
- -x^3+4*x^2-15*x-30=0
- x^3+4*x^2-15*x+30=0
-x^3+4*x^2-15*x+30=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 - x + 4*x - 15*x + 30 = 0
$$\left(- 15 x + \left(- x^{3} + 4 x^{2}\right)\right) + 30 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 15 x + \left(- x^{3} + 4 x^{2}\right)\right) + 30 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x^{2} + 15 x - 30 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 15$$
$$v = \frac{d}{a}$$
$$v = -30$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 15$$
$$x_{1} x_{2} x_{3} = -30$$
$$\left(- 15 x + \left(- x^{3} + 4 x^{2}\right)\right) + 30 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x^{2} + 15 x - 30 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 15$$
$$v = \frac{d}{a}$$
$$v = -30$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 15$$
$$x_{1} x_{2} x_{3} = -30$$
Respuesta rápida
[src]
_______________ / _______________\
/ _____ | / _____ |
/ 199 \/ 790 | ___ / 199 \/ 790 |
3 / --- + ------- | ___ \/ 3 *3 / --- + ------- |
4 \/ 27 3 29 | 29*\/ 3 \/ 27 3 |
x1 = - - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------|
3 2 _______________ | _______________ 2 |
/ _____ | / _____ |
/ 199 \/ 790 | / 199 \/ 790 |
18*3 / --- + ------- | 18*3 / --- + ------- |
\/ 27 3 \ \/ 27 3 /
$$x_{1} = - \frac{\sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} + \frac{29}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}\right)$$
_______________ / _______________ \
/ _____ | / _____ |
/ 199 \/ 790 | ___ / 199 \/ 790 |
3 / --- + ------- |\/ 3 *3 / --- + ------- ___ |
4 \/ 27 3 29 | \/ 27 3 29*\/ 3 |
x2 = - - -------------------- + ----------------------- + I*|-------------------------- + -----------------------|
3 2 _______________ | 2 _______________|
/ _____ | / _____ |
/ 199 \/ 790 | / 199 \/ 790 |
18*3 / --- + ------- | 18*3 / --- + ------- |
\/ 27 3 \ \/ 27 3 /
$$x_{2} = - \frac{\sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} + \frac{29}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2}\right)$$
_______________
/ _____
4 / 199 \/ 790 29
x3 = - + 3 / --- + ------- - ----------------------
3 \/ 27 3 _______________
/ _____
/ 199 \/ 790
9*3 / --- + -------
\/ 27 3
$$x_{3} = - \frac{29}{9 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}$$
x3 = -29/(9*(199/27 + sqrt(790)/3)^(1/3)) + 4/3 + (199/27 + sqrt(790)/3)^(1/3)
Suma y producto de raíces
[src]
suma
_______________ / _______________\ _______________ / _______________ \
/ _____ | / _____ | / _____ | / _____ |
/ 199 \/ 790 | ___ / 199 \/ 790 | / 199 \/ 790 | ___ / 199 \/ 790 | _______________
3 / --- + ------- | ___ \/ 3 *3 / --- + ------- | 3 / --- + ------- |\/ 3 *3 / --- + ------- ___ | / _____
4 \/ 27 3 29 | 29*\/ 3 \/ 27 3 | 4 \/ 27 3 29 | \/ 27 3 29*\/ 3 | 4 / 199 \/ 790 29
- - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------| + - - -------------------- + ----------------------- + I*|-------------------------- + -----------------------| + - + 3 / --- + ------- - ----------------------
3 2 _______________ | _______________ 2 | 3 2 _______________ | 2 _______________| 3 \/ 27 3 _______________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 199 \/ 790 | / 199 \/ 790 | / 199 \/ 790 | / 199 \/ 790 | / 199 \/ 790
18*3 / --- + ------- | 18*3 / --- + ------- | 18*3 / --- + ------- | 18*3 / --- + ------- | 9*3 / --- + -------
\/ 27 3 \ \/ 27 3 / \/ 27 3 \ \/ 27 3 / \/ 27 3
$$\left(- \frac{29}{9 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} + \frac{29}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} + \frac{29}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2}\right)\right)\right)$$
=
/ _______________ \ / _______________\
| / _____ | | / _____ |
| ___ / 199 \/ 790 | | ___ / 199 \/ 790 |
|\/ 3 *3 / --- + ------- ___ | | ___ \/ 3 *3 / --- + ------- |
| \/ 27 3 29*\/ 3 | | 29*\/ 3 \/ 27 3 |
4 + I*|-------------------------- + -----------------------| + I*|- ----------------------- - --------------------------|
| 2 _______________| | _______________ 2 |
| / _____ | | / _____ |
| / 199 \/ 790 | | / 199 \/ 790 |
| 18*3 / --- + ------- | | 18*3 / --- + ------- |
\ \/ 27 3 / \ \/ 27 3 /
$$4 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}\right) + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2}\right)$$
producto
/ _______________ / _______________\\ / _______________ / _______________ \\ | / _____ | / _____ || | / _____ | / _____ || | / 199 \/ 790 | ___ / 199 \/ 790 || | / 199 \/ 790 | ___ / 199 \/ 790 || / _______________ \ | 3 / --- + ------- | ___ \/ 3 *3 / --- + ------- || | 3 / --- + ------- |\/ 3 *3 / --- + ------- ___ || | / _____ | |4 \/ 27 3 29 | 29*\/ 3 \/ 27 3 || |4 \/ 27 3 29 | \/ 27 3 29*\/ 3 || |4 / 199 \/ 790 29 | |- - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------||*|- - -------------------- + ----------------------- + I*|-------------------------- + -----------------------||*|- + 3 / --- + ------- - ----------------------| |3 2 _______________ | _______________ 2 || |3 2 _______________ | 2 _______________|| |3 \/ 27 3 _______________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 199 \/ 790 | / 199 \/ 790 || | / 199 \/ 790 | / 199 \/ 790 || | / 199 \/ 790 | | 18*3 / --- + ------- | 18*3 / --- + ------- || | 18*3 / --- + ------- | 18*3 / --- + ------- || | 9*3 / --- + ------- | \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 /
$$\left(- \frac{\sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} + \frac{29}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} + \frac{29}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}}\right)\right) \left(- \frac{29}{9 \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}} + \frac{4}{3} + \sqrt[3]{\frac{199}{27} + \frac{\sqrt{790}}{3}}\right)$$
=
30
$$30$$
30
Respuesta numérica
[src]
x1 = 2.63177698620825
x2 = 0.684111506895875 + 3.3062264904926*i
x3 = 0.684111506895875 - 3.3062264904926*i
x3 = 0.684111506895875 - 3.3062264904926*i