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Expresiones idénticas
- x^ tres - cuarenta y nueve *x- ciento treinta y ocho = cero
- x al cubo menos 49 multiplicar por x menos 138 es igual a 0
- x en el grado tres menos cuarenta y nueve multiplicar por x menos ciento treinta y ocho es igual a cero
- x3-49*x-138=0
- x³-49*x-138=0
- x en el grado 3-49*x-138=0
- x^3-49x-138=0
- x3-49x-138=0
- x^3-49*x-138=O
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Expresiones semejantes
- x^3+49*x-138=0
- x^3-49*x+138=0
x^3-49*x-138=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -49$$
$$v = \frac{d}{a}$$
$$v = -138$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -49$$
$$x_{1} x_{2} x_{3} = -138$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -49$$
$$v = \frac{d}{a}$$
$$v = -138$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -49$$
$$x_{1} x_{2} x_{3} = -138$$
Suma y producto de raíces
[src]
suma
________________ / ________________ \ ________________ / ________________ \
/ _______ | / _______ | / _______ | / _______ |
/ \/ 32694 | ___ / \/ 32694 | / \/ 32694 | ___ / \/ 32694 | ________________
3 / 69 + --------- | \/ 3 *3 / 69 + --------- ___ | 3 / 69 + --------- |\/ 3 *3 / 69 + --------- ___ | / _______
49 \/ 9 | \/ 9 49*\/ 3 | 49 \/ 9 | \/ 9 49*\/ 3 | / \/ 32694 49
- ----------------------- - --------------------- + I*|- --------------------------- + -----------------------| + - ----------------------- - --------------------- + I*|--------------------------- - -----------------------| + 3 / 69 + --------- + -----------------------
________________ 2 | 2 ________________| ________________ 2 | 2 ________________| \/ 9 ________________
/ _______ | / _______ | / _______ | / _______ | / _______
/ \/ 32694 | / \/ 32694 | / \/ 32694 | / \/ 32694 | / \/ 32694
6*3 / 69 + --------- | 6*3 / 69 + --------- | 6*3 / 69 + --------- | 6*3 / 69 + --------- | 3*3 / 69 + ---------
\/ 9 \ \/ 9 / \/ 9 \ \/ 9 / \/ 9
$$\left(\frac{49}{3 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} - \frac{49}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} + \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} - \frac{49}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + i \left(- \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________ \ | / _______ | | / _______ | | ___ / \/ 32694 | | ___ / \/ 32694 | |\/ 3 *3 / 69 + --------- ___ | | \/ 3 *3 / 69 + --------- ___ | | \/ 9 49*\/ 3 | | \/ 9 49*\/ 3 | I*|--------------------------- - -----------------------| + I*|- --------------------------- + -----------------------| | 2 ________________| | 2 ________________| | / _______ | | / _______ | | / \/ 32694 | | / \/ 32694 | | 6*3 / 69 + --------- | | 6*3 / 69 + --------- | \ \/ 9 / \ \/ 9 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} + \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}\right) + i \left(- \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2}\right)$$
producto
/ ________________ / ________________ \\ / ________________ / ________________ \\ | / _______ | / _______ || | / _______ | / _______ || | / \/ 32694 | ___ / \/ 32694 || | / \/ 32694 | ___ / \/ 32694 || / ________________ \ | 3 / 69 + --------- | \/ 3 *3 / 69 + --------- ___ || | 3 / 69 + --------- |\/ 3 *3 / 69 + --------- ___ || | / _______ | | 49 \/ 9 | \/ 9 49*\/ 3 || | 49 \/ 9 | \/ 9 49*\/ 3 || | / \/ 32694 49 | |- ----------------------- - --------------------- + I*|- --------------------------- + -----------------------||*|- ----------------------- - --------------------- + I*|--------------------------- - -----------------------||*|3 / 69 + --------- + -----------------------| | ________________ 2 | 2 ________________|| | ________________ 2 | 2 ________________|| |\/ 9 ________________| | / _______ | / _______ || | / _______ | / _______ || | / _______ | | / \/ 32694 | / \/ 32694 || | / \/ 32694 | / \/ 32694 || | / \/ 32694 | | 6*3 / 69 + --------- | 6*3 / 69 + --------- || | 6*3 / 69 + --------- | 6*3 / 69 + --------- || | 3*3 / 69 + --------- | \ \/ 9 \ \/ 9 // \ \/ 9 \ \/ 9 // \ \/ 9 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} - \frac{49}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + i \left(- \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} - \frac{49}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} + \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}\right)\right) \left(\frac{49}{3 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}\right)$$
=
138
$$138$$
138
Respuesta rápida
[src]
________________ / ________________ \
/ _______ | / _______ |
/ \/ 32694 | ___ / \/ 32694 |
3 / 69 + --------- | \/ 3 *3 / 69 + --------- ___ |
49 \/ 9 | \/ 9 49*\/ 3 |
x1 = - ----------------------- - --------------------- + I*|- --------------------------- + -----------------------|
________________ 2 | 2 ________________|
/ _______ | / _______ |
/ \/ 32694 | / \/ 32694 |
6*3 / 69 + --------- | 6*3 / 69 + --------- |
\/ 9 \ \/ 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} - \frac{49}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} + \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}\right)$$
________________ / ________________ \
/ _______ | / _______ |
/ \/ 32694 | ___ / \/ 32694 |
3 / 69 + --------- |\/ 3 *3 / 69 + --------- ___ |
49 \/ 9 | \/ 9 49*\/ 3 |
x2 = - ----------------------- - --------------------- + I*|--------------------------- - -----------------------|
________________ 2 | 2 ________________|
/ _______ | / _______ |
/ \/ 32694 | / \/ 32694 |
6*3 / 69 + --------- | 6*3 / 69 + --------- |
\/ 9 \ \/ 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2} - \frac{49}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + i \left(- \frac{49 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}}{2}\right)$$
________________
/ _______
/ \/ 32694 49
x3 = 3 / 69 + --------- + -----------------------
\/ 9 ________________
/ _______
/ \/ 32694
3*3 / 69 + ---------
\/ 9
$$x_{3} = \frac{49}{3 \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}} + \sqrt[3]{\frac{\sqrt{32694}}{9} + 69}$$
x3 = 49/(3*(sqrt(32694)/9 + 69)^(1/3)) + (sqrt(32694)/9 + 69)^(1/3)
Respuesta numérica
[src]
x1 = 8.12330910368656
x2 = -4.06165455184328 + 0.700794617222248*i
x3 = -4.06165455184328 - 0.700794617222248*i
x3 = -4.06165455184328 - 0.700794617222248*i