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Expresiones semejantes
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- -1*x^3+5*x^2-12*x-15=0
-1*x^3+5*x^2-12*x+15=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 - x + 5*x - 12*x + 15 = 0
$$\left(- 12 x + \left(- x^{3} + 5 x^{2}\right)\right) + 15 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 12 x + \left(- x^{3} + 5 x^{2}\right)\right) + 15 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 5 x^{2} + 12 x - 15 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 12$$
$$v = \frac{d}{a}$$
$$v = -15$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 12$$
$$x_{1} x_{2} x_{3} = -15$$
$$\left(- 12 x + \left(- x^{3} + 5 x^{2}\right)\right) + 15 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 5 x^{2} + 12 x - 15 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 12$$
$$v = \frac{d}{a}$$
$$v = -15$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 12$$
$$x_{1} x_{2} x_{3} = -15$$
Respuesta rápida
[src]
_______________ / _______________\
/ _____ | / _____ |
/ 115 \/ 229 | ___ / 115 \/ 229 |
3 / --- + ------- | ___ \/ 3 *3 / --- + ------- |
5 \/ 54 6 11 | 11*\/ 3 \/ 54 6 |
x1 = - - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------|
3 2 _______________ | _______________ 2 |
/ _____ | / _____ |
/ 115 \/ 229 | / 115 \/ 229 |
18*3 / --- + ------- | 18*3 / --- + ------- |
\/ 54 6 \ \/ 54 6 /
$$x_{1} = - \frac{\sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} + \frac{11}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} - \frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}\right)$$
_______________ / _______________ \
/ _____ | / _____ |
/ 115 \/ 229 | ___ / 115 \/ 229 |
3 / --- + ------- |\/ 3 *3 / --- + ------- ___ |
5 \/ 54 6 11 | \/ 54 6 11*\/ 3 |
x2 = - - -------------------- + ----------------------- + I*|-------------------------- + -----------------------|
3 2 _______________ | 2 _______________|
/ _____ | / _____ |
/ 115 \/ 229 | / 115 \/ 229 |
18*3 / --- + ------- | 18*3 / --- + ------- |
\/ 54 6 \ \/ 54 6 /
$$x_{2} = - \frac{\sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} + \frac{11}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + i \left(\frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2}\right)$$
_______________
/ _____
5 / 115 \/ 229 11
x3 = - + 3 / --- + ------- - ----------------------
3 \/ 54 6 _______________
/ _____
/ 115 \/ 229
9*3 / --- + -------
\/ 54 6
$$x_{3} = - \frac{11}{9 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}$$
x3 = -11/(9*(115/54 + sqrt(229)/6)^(1/3)) + 5/3 + (115/54 + sqrt(229)/6)^(1/3)
Suma y producto de raíces
[src]
suma
_______________ / _______________\ _______________ / _______________ \
/ _____ | / _____ | / _____ | / _____ |
/ 115 \/ 229 | ___ / 115 \/ 229 | / 115 \/ 229 | ___ / 115 \/ 229 | _______________
3 / --- + ------- | ___ \/ 3 *3 / --- + ------- | 3 / --- + ------- |\/ 3 *3 / --- + ------- ___ | / _____
5 \/ 54 6 11 | 11*\/ 3 \/ 54 6 | 5 \/ 54 6 11 | \/ 54 6 11*\/ 3 | 5 / 115 \/ 229 11
- - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------| + - - -------------------- + ----------------------- + I*|-------------------------- + -----------------------| + - + 3 / --- + ------- - ----------------------
3 2 _______________ | _______________ 2 | 3 2 _______________ | 2 _______________| 3 \/ 54 6 _______________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 115 \/ 229 | / 115 \/ 229 | / 115 \/ 229 | / 115 \/ 229 | / 115 \/ 229
18*3 / --- + ------- | 18*3 / --- + ------- | 18*3 / --- + ------- | 18*3 / --- + ------- | 9*3 / --- + -------
\/ 54 6 \ \/ 54 6 / \/ 54 6 \ \/ 54 6 / \/ 54 6
$$\left(- \frac{11}{9 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} + \frac{11}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} - \frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} + \frac{11}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + i \left(\frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2}\right)\right)\right)$$
=
/ _______________ \ / _______________\
| / _____ | | / _____ |
| ___ / 115 \/ 229 | | ___ / 115 \/ 229 |
|\/ 3 *3 / --- + ------- ___ | | ___ \/ 3 *3 / --- + ------- |
| \/ 54 6 11*\/ 3 | | 11*\/ 3 \/ 54 6 |
5 + I*|-------------------------- + -----------------------| + I*|- ----------------------- - --------------------------|
| 2 _______________| | _______________ 2 |
| / _____ | | / _____ |
| / 115 \/ 229 | | / 115 \/ 229 |
| 18*3 / --- + ------- | | 18*3 / --- + ------- |
\ \/ 54 6 / \ \/ 54 6 /
$$5 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} - \frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}\right) + i \left(\frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2}\right)$$
producto
/ _______________ / _______________\\ / _______________ / _______________ \\ | / _____ | / _____ || | / _____ | / _____ || | / 115 \/ 229 | ___ / 115 \/ 229 || | / 115 \/ 229 | ___ / 115 \/ 229 || / _______________ \ | 3 / --- + ------- | ___ \/ 3 *3 / --- + ------- || | 3 / --- + ------- |\/ 3 *3 / --- + ------- ___ || | / _____ | |5 \/ 54 6 11 | 11*\/ 3 \/ 54 6 || |5 \/ 54 6 11 | \/ 54 6 11*\/ 3 || |5 / 115 \/ 229 11 | |- - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------||*|- - -------------------- + ----------------------- + I*|-------------------------- + -----------------------||*|- + 3 / --- + ------- - ----------------------| |3 2 _______________ | _______________ 2 || |3 2 _______________ | 2 _______________|| |3 \/ 54 6 _______________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 115 \/ 229 | / 115 \/ 229 || | / 115 \/ 229 | / 115 \/ 229 || | / 115 \/ 229 | | 18*3 / --- + ------- | 18*3 / --- + ------- || | 18*3 / --- + ------- | 18*3 / --- + ------- || | 9*3 / --- + ------- | \ \/ 54 6 \ \/ 54 6 // \ \/ 54 6 \ \/ 54 6 // \ \/ 54 6 /
$$\left(- \frac{\sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} + \frac{11}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + i \left(\frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} + \frac{11}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}{2} - \frac{11 \sqrt{3}}{18 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}}\right)\right) \left(- \frac{11}{9 \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}} + \frac{5}{3} + \sqrt[3]{\frac{115}{54} + \frac{\sqrt{229}}{6}}\right)$$
=
15
$$15$$
15
Respuesta numérica
[src]
x1 = 1.1980924212362 + 2.07974809462801*i
x2 = 1.1980924212362 - 2.07974809462801*i
x3 = 2.60381515752759
x3 = 2.60381515752759