Sr Examen
x^3+3x-5=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = -5$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = -5$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = -5$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = -5$$
Respuesta rápida
[src]
____________ / ____________\
/ ____ | / ____ |
/ 5 \/ 29 | ___ / 5 \/ 29 |
3 / - + ------ | ___ \/ 3 *3 / - + ------ |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = ------------------- - ----------------- + I*|- ------------------- - -----------------------|
____________ 2 | ____________ 2 |
/ ____ | / ____ |
/ 5 \/ 29 | / 5 \/ 29 |
2*3 / - + ------ | 2*3 / - + ------ |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} + \frac{1}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}\right)$$
____________ / ____________\
/ ____ | / ____ |
/ 5 \/ 29 | ___ / 5 \/ 29 |
3 / - + ------ | ___ \/ 3 *3 / - + ------ |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x2 = ------------------- - ----------------- + I*|------------------- + -----------------------|
____________ 2 | ____________ 2 |
/ ____ | / ____ |
/ 5 \/ 29 | / 5 \/ 29 |
2*3 / - + ------ |2*3 / - + ------ |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} + \frac{1}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2}\right)$$
____________
/ ____
/ 5 \/ 29 1
x3 = 3 / - + ------ - -----------------
\/ 2 2 ____________
/ ____
/ 5 \/ 29
3 / - + ------
\/ 2 2
$$x_{3} = - \frac{1}{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}$$
x3 = -1/(5/2 + sqrt(29)/2)^(1/3) + (5/2 + sqrt(29)/2)^(1/3)
Suma y producto de raíces
[src]
suma
____________ / ____________\ ____________ / ____________\
/ ____ | / ____ | / ____ | / ____ |
/ 5 \/ 29 | ___ / 5 \/ 29 | / 5 \/ 29 | ___ / 5 \/ 29 | ____________
3 / - + ------ | ___ \/ 3 *3 / - + ------ | 3 / - + ------ | ___ \/ 3 *3 / - + ------ | / ____
1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2 | \/ 3 \/ 2 2 | / 5 \/ 29 1
------------------- - ----------------- + I*|- ------------------- - -----------------------| + ------------------- - ----------------- + I*|------------------- + -----------------------| + 3 / - + ------ - -----------------
____________ 2 | ____________ 2 | ____________ 2 | ____________ 2 | \/ 2 2 ____________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 5 \/ 29 | / 5 \/ 29 | / 5 \/ 29 | / 5 \/ 29 | / 5 \/ 29
2*3 / - + ------ | 2*3 / - + ------ | 2*3 / - + ------ |2*3 / - + ------ | 3 / - + ------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{1}{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} + \frac{1}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} + \frac{1}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2}\right)\right)\right)$$
=
/ ____________\ / ____________\ | / ____ | | / ____ | | ___ / 5 \/ 29 | | ___ / 5 \/ 29 | | ___ \/ 3 *3 / - + ------ | | ___ \/ 3 *3 / - + ------ | | \/ 3 \/ 2 2 | | \/ 3 \/ 2 2 | I*|------------------- + -----------------------| + I*|- ------------------- - -----------------------| | ____________ 2 | | ____________ 2 | | / ____ | | / ____ | | / 5 \/ 29 | | / 5 \/ 29 | |2*3 / - + ------ | | 2*3 / - + ------ | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2}\right)$$
producto
/ ____________ / ____________\\ / ____________ / ____________\\ | / ____ | / ____ || | / ____ | / ____ || | / 5 \/ 29 | ___ / 5 \/ 29 || | / 5 \/ 29 | ___ / 5 \/ 29 || / ____________ \ | 3 / - + ------ | ___ \/ 3 *3 / - + ------ || | 3 / - + ------ | ___ \/ 3 *3 / - + ------ || | / ____ | | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | \/ 3 \/ 2 2 || | / 5 \/ 29 1 | |------------------- - ----------------- + I*|- ------------------- - -----------------------||*|------------------- - ----------------- + I*|------------------- + -----------------------||*|3 / - + ------ - -----------------| | ____________ 2 | ____________ 2 || | ____________ 2 | ____________ 2 || |\/ 2 2 ____________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 5 \/ 29 | / 5 \/ 29 || | / 5 \/ 29 | / 5 \/ 29 || | / 5 \/ 29 | |2*3 / - + ------ | 2*3 / - + ------ || |2*3 / - + ------ |2*3 / - + ------ || | 3 / - + ------ | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(- \frac{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} + \frac{1}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} + \frac{1}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}{2} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}}\right)\right) \left(- \frac{1}{\sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}} + \sqrt[3]{\frac{5}{2} + \frac{\sqrt{29}}{2}}\right)$$
=
5
$$5$$
5
Respuesta numérica
[src]
x1 = 1.15417149518144
x2 = -0.577085747590721 + 1.99977095693907*i
x3 = -0.577085747590721 - 1.99977095693907*i
x3 = -0.577085747590721 - 1.99977095693907*i