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- La ecuación:
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Ecuación x=(x+1)/2
- Ecuación (x-6)(-5x-9)=0
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Ecuación x^2-14*x+33=0
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Ecuación x^2-5x=14
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Expresiones idénticas
- x^ tres -34x-175x+ diez mil = cero
- x al cubo menos 34x menos 175x más 10000 es igual a 0
- x en el grado tres menos 34x menos 175x más diez mil es igual a cero
- x3-34x-175x+10000=0
- x³-34x-175x+10000=0
- x en el grado 3-34x-175x+10000=0
- x^3-34x-175x+10000=O
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Expresiones semejantes
- x^3+34x-175x+10000=0
- x^3-34x+175x+10000=0
- x^3-34x-175x-10000=0
x^3-34x-175x+10000=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
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3 x - 34*x - 175*x + 10000 = 0
$$\left(- 175 x + \left(x^{3} - 34 x\right)\right) + 10000 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -209$$
$$v = \frac{d}{a}$$
$$v = 10000$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -209$$
$$x_{1} x_{2} x_{3} = 10000$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -209$$
$$v = \frac{d}{a}$$
$$v = 10000$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -209$$
$$x_{1} x_{2} x_{3} = 10000$$
Respuesta rápida
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___________________________ / ___________________________\
3 / ____________ | ___ ___ 3 / ____________ |
\/ 135000 + 3*\/ 1997612013 209 | 209*\/ 3 \/ 3 *\/ 135000 + 3*\/ 1997612013 |
x1 = ------------------------------ + -------------------------------- + I*|- -------------------------------- + ------------------------------------|
6 ___________________________ | ___________________________ 6 |
3 / ____________ | 3 / ____________ |
2*\/ 135000 + 3*\/ 1997612013 \ 2*\/ 135000 + 3*\/ 1997612013 /
$$x_{1} = \frac{209}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + i \left(- \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6}\right)$$
___________________________ / ___________________________ \
3 / ____________ | ___ 3 / ____________ ___ |
\/ 135000 + 3*\/ 1997612013 209 | \/ 3 *\/ 135000 + 3*\/ 1997612013 209*\/ 3 |
x2 = ------------------------------ + -------------------------------- + I*|- ------------------------------------ + --------------------------------|
6 ___________________________ | 6 ___________________________|
3 / ____________ | 3 / ____________ |
2*\/ 135000 + 3*\/ 1997612013 \ 2*\/ 135000 + 3*\/ 1997612013 /
$$x_{2} = \frac{209}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}}\right)$$
___________________________
3 / ____________
209 \/ 135000 + 3*\/ 1997612013
x3 = - ------------------------------ - ------------------------------
___________________________ 3
3 / ____________
\/ 135000 + 3*\/ 1997612013
$$x_{3} = - \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{3} - \frac{209}{\sqrt[3]{3 \sqrt{1997612013} + 135000}}$$
x3 = -(3*sqrt(1997612013) + 135000)^(1/3)/3 - 209/(3*sqrt(1997612013) + 135000)^(1/3)
Suma y producto de raíces
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suma
___________________________ / ___________________________\ ___________________________ / ___________________________ \ ___________________________
3 / ____________ | ___ ___ 3 / ____________ | 3 / ____________ | ___ 3 / ____________ ___ | 3 / ____________
\/ 135000 + 3*\/ 1997612013 209 | 209*\/ 3 \/ 3 *\/ 135000 + 3*\/ 1997612013 | \/ 135000 + 3*\/ 1997612013 209 | \/ 3 *\/ 135000 + 3*\/ 1997612013 209*\/ 3 | 209 \/ 135000 + 3*\/ 1997612013
------------------------------ + -------------------------------- + I*|- -------------------------------- + ------------------------------------| + ------------------------------ + -------------------------------- + I*|- ------------------------------------ + --------------------------------| + - ------------------------------ - ------------------------------
6 ___________________________ | ___________________________ 6 | 6 ___________________________ | 6 ___________________________| ___________________________ 3
3 / ____________ | 3 / ____________ | 3 / ____________ | 3 / ____________ | 3 / ____________
2*\/ 135000 + 3*\/ 1997612013 \ 2*\/ 135000 + 3*\/ 1997612013 / 2*\/ 135000 + 3*\/ 1997612013 \ 2*\/ 135000 + 3*\/ 1997612013 / \/ 135000 + 3*\/ 1997612013
$$\left(- \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{3} - \frac{209}{\sqrt[3]{3 \sqrt{1997612013} + 135000}}\right) + \left(\left(\frac{209}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}}\right)\right) + \left(\frac{209}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + i \left(- \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6}\right)\right)\right)$$
=
/ ___________________________\ / ___________________________ \ | ___ ___ 3 / ____________ | | ___ 3 / ____________ ___ | | 209*\/ 3 \/ 3 *\/ 135000 + 3*\/ 1997612013 | | \/ 3 *\/ 135000 + 3*\/ 1997612013 209*\/ 3 | I*|- -------------------------------- + ------------------------------------| + I*|- ------------------------------------ + --------------------------------| | ___________________________ 6 | | 6 ___________________________| | 3 / ____________ | | 3 / ____________ | \ 2*\/ 135000 + 3*\/ 1997612013 / \ 2*\/ 135000 + 3*\/ 1997612013 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}}\right) + i \left(- \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6}\right)$$
producto
/ ___________________________ / ___________________________\\ / ___________________________ / ___________________________ \\ / ___________________________\ |3 / ____________ | ___ ___ 3 / ____________ || |3 / ____________ | ___ 3 / ____________ ___ || | 3 / ____________ | |\/ 135000 + 3*\/ 1997612013 209 | 209*\/ 3 \/ 3 *\/ 135000 + 3*\/ 1997612013 || |\/ 135000 + 3*\/ 1997612013 209 | \/ 3 *\/ 135000 + 3*\/ 1997612013 209*\/ 3 || | 209 \/ 135000 + 3*\/ 1997612013 | |------------------------------ + -------------------------------- + I*|- -------------------------------- + ------------------------------------||*|------------------------------ + -------------------------------- + I*|- ------------------------------------ + --------------------------------||*|- ------------------------------ - ------------------------------| | 6 ___________________________ | ___________________________ 6 || | 6 ___________________________ | 6 ___________________________|| | ___________________________ 3 | | 3 / ____________ | 3 / ____________ || | 3 / ____________ | 3 / ____________ || | 3 / ____________ | \ 2*\/ 135000 + 3*\/ 1997612013 \ 2*\/ 135000 + 3*\/ 1997612013 // \ 2*\/ 135000 + 3*\/ 1997612013 \ 2*\/ 135000 + 3*\/ 1997612013 // \ \/ 135000 + 3*\/ 1997612013 /
$$\left(\frac{209}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + i \left(- \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6}\right)\right) \left(\frac{209}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}} + \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{1997612013} + 135000}}{6} + \frac{209 \sqrt{3}}{2 \sqrt[3]{3 \sqrt{1997612013} + 135000}}\right)\right) \left(- \frac{\sqrt[3]{3 \sqrt{1997612013} + 135000}}{3} - \frac{209}{\sqrt[3]{3 \sqrt{1997612013} + 135000}}\right)$$
=
-10000
$$-10000$$
-10000
Respuesta numérica
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x1 = -24.7572599630794
x2 = 12.3786299815397 + 15.8332384766863*i
x3 = 12.3786299815397 - 15.8332384766863*i
x3 = 12.3786299815397 - 15.8332384766863*i