Sr Examen
X^3-4x^2-9x-36=0 la ecuación
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Solución
Ha introducido
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3 2 x - 4*x - 9*x - 36 = 0
$$\left(- 9 x + \left(x^{3} - 4 x^{2}\right)\right) - 36 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -36$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -36$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -36$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -36$$
Suma y producto de raíces
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suma
________________ / ________________ \ ________________ / ________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 712 \/ 5277 | ___ / 712 \/ 5277 | / 712 \/ 5277 | ___ / 712 \/ 5277 | ________________
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ | / ______
4 43 \/ 27 3 | \/ 27 3 43*\/ 3 | 4 43 \/ 27 3 | \/ 27 3 43*\/ 3 | 4 / 712 \/ 5277 43
- - ------------------------ - --------------------- + I*|- --------------------------- + ------------------------| + - - ------------------------ - --------------------- + I*|--------------------------- - ------------------------| + - + 3 / --- + -------- + -----------------------
3 ________________ 2 | 2 ________________| 3 ________________ 2 | 2 ________________| 3 \/ 27 3 ________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 712 \/ 5277 | / 712 \/ 5277 | / 712 \/ 5277 | / 712 \/ 5277 | / 712 \/ 5277
18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 9*3 / --- + --------
\/ 27 3 \ \/ 27 3 / \/ 27 3 \ \/ 27 3 / \/ 27 3
$$\left(\frac{43}{9 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} - \frac{43}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} + \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} - \frac{43}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + i \left(- \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________ \
| / ______ | | / ______ |
| ___ / 712 \/ 5277 | | ___ / 712 \/ 5277 |
|\/ 3 *3 / --- + -------- ___ | | \/ 3 *3 / --- + -------- ___ |
| \/ 27 3 43*\/ 3 | | \/ 27 3 43*\/ 3 |
4 + I*|--------------------------- - ------------------------| + I*|- --------------------------- + ------------------------|
| 2 ________________| | 2 ________________|
| / ______ | | / ______ |
| / 712 \/ 5277 | | / 712 \/ 5277 |
| 18*3 / --- + -------- | | 18*3 / --- + -------- |
\ \/ 27 3 / \ \/ 27 3 /
$$4 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} + \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}\right) + i \left(- \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2}\right)$$
producto
/ ________________ / ________________ \\ / ________________ / ________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 712 \/ 5277 | ___ / 712 \/ 5277 || | / 712 \/ 5277 | ___ / 712 \/ 5277 || / ________________ \ | 3 / --- + -------- | \/ 3 *3 / --- + -------- ___ || | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ || | / ______ | |4 43 \/ 27 3 | \/ 27 3 43*\/ 3 || |4 43 \/ 27 3 | \/ 27 3 43*\/ 3 || |4 / 712 \/ 5277 43 | |- - ------------------------ - --------------------- + I*|- --------------------------- + ------------------------||*|- - ------------------------ - --------------------- + I*|--------------------------- - ------------------------||*|- + 3 / --- + -------- + -----------------------| |3 ________________ 2 | 2 ________________|| |3 ________________ 2 | 2 ________________|| |3 \/ 27 3 ________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 712 \/ 5277 | / 712 \/ 5277 || | / 712 \/ 5277 | / 712 \/ 5277 || | / 712 \/ 5277 | | 18*3 / --- + -------- | 18*3 / --- + -------- || | 18*3 / --- + -------- | 18*3 / --- + -------- || | 9*3 / --- + -------- | \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} - \frac{43}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + i \left(- \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} - \frac{43}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} + \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}\right)\right) \left(\frac{43}{9 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}\right)$$
=
36
$$36$$
36
Respuesta rápida
[src]
________________ / ________________ \
/ ______ | / ______ |
/ 712 \/ 5277 | ___ / 712 \/ 5277 |
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ |
4 43 \/ 27 3 | \/ 27 3 43*\/ 3 |
x1 = - - ------------------------ - --------------------- + I*|- --------------------------- + ------------------------|
3 ________________ 2 | 2 ________________|
/ ______ | / ______ |
/ 712 \/ 5277 | / 712 \/ 5277 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 27 3 \ \/ 27 3 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} - \frac{43}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} + \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}\right)$$
________________ / ________________ \
/ ______ | / ______ |
/ 712 \/ 5277 | ___ / 712 \/ 5277 |
3 / --- + -------- |\/ 3 *3 / --- + -------- ___ |
4 43 \/ 27 3 | \/ 27 3 43*\/ 3 |
x2 = - - ------------------------ - --------------------- + I*|--------------------------- - ------------------------|
3 ________________ 2 | 2 ________________|
/ ______ | / ______ |
/ 712 \/ 5277 | / 712 \/ 5277 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 27 3 \ \/ 27 3 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2} - \frac{43}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + i \left(- \frac{43 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}}{2}\right)$$
________________
/ ______
4 / 712 \/ 5277 43
x3 = - + 3 / --- + -------- + -----------------------
3 \/ 27 3 ________________
/ ______
/ 712 \/ 5277
9*3 / --- + --------
\/ 27 3
$$x_{3} = \frac{43}{9 \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}} + \frac{4}{3} + \sqrt[3]{\frac{\sqrt{5277}}{3} + \frac{712}{27}}$$
x3 = 43/(9*(sqrt(5277)/3 + 712/27)^(1/3)) + 4/3 + (sqrt(5277)/3 + 712/27)^(1/3)
Respuesta numérica
[src]
x1 = -1.16177081654556 + 2.08405873787071*i
x2 = 6.32354163309112
x3 = -1.16177081654556 - 2.08405873787071*i
x3 = -1.16177081654556 - 2.08405873787071*i