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Ecuación 2*x^2+6*x+9=
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Expresiones idénticas
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- -x3+12x2-6x+8=0
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- -x^3+12x^2-6x+8=O
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Expresiones semejantes
- x^3+12x^2-6x+8=0
- -x^3+12x^2+6x+8=0
- -x^3+12x^2-6x-8=0
- -x^3-12x^2-6x+8=0
-x^3+12x^2-6x+8=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 - x + 12*x - 6*x + 8 = 0
$$\left(- 6 x + \left(- x^{3} + 12 x^{2}\right)\right) + 8 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 6 x + \left(- x^{3} + 12 x^{2}\right)\right) + 8 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 12 x^{2} + 6 x - 8 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -12$$
$$q = \frac{c}{a}$$
$$q = 6$$
$$v = \frac{d}{a}$$
$$v = -8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 12$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 6$$
$$x_{1} x_{2} x_{3} = -8$$
$$\left(- 6 x + \left(- x^{3} + 12 x^{2}\right)\right) + 8 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 12 x^{2} + 6 x - 8 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -12$$
$$q = \frac{c}{a}$$
$$q = 6$$
$$v = \frac{d}{a}$$
$$v = -8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 12$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 6$$
$$x_{1} x_{2} x_{3} = -8$$
Respuesta rápida
[src]
_______________ / _______________\
3 / ___ | ___ ___ 3 / ___ |
7 \/ 56 + 14*\/ 2 | 7*\/ 3 \/ 3 *\/ 56 + 14*\/ 2 |
x1 = 4 - ------------------ - ------------------ + I*|------------------ - ------------------------|
_______________ 2 | _______________ 2 |
3 / ___ |3 / ___ |
\/ 56 + 14*\/ 2 \\/ 56 + 14*\/ 2 /
$$x_{1} = - \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right)$$
_______________ / _______________ \
3 / ___ | ___ 3 / ___ ___ |
7 \/ 56 + 14*\/ 2 |\/ 3 *\/ 56 + 14*\/ 2 7*\/ 3 |
x2 = 4 - ------------------ - ------------------ + I*|------------------------ - ------------------|
_______________ 2 | 2 _______________|
3 / ___ | 3 / ___ |
\/ 56 + 14*\/ 2 \ \/ 56 + 14*\/ 2 /
$$x_{2} = - \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right)$$
_______________
3 / ___ 14
x3 = 4 + \/ 56 + 14*\/ 2 + ------------------
_______________
3 / ___
\/ 56 + 14*\/ 2
$$x_{3} = \frac{14}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + \sqrt[3]{14 \sqrt{2} + 56}$$
x3 = 14/(14*sqrt(2) + 56)^(1/3) + 4 + (14*sqrt(2) + 56)^(1/3)
Suma y producto de raíces
[src]
suma
_______________ / _______________\ _______________ / _______________ \
3 / ___ | ___ ___ 3 / ___ | 3 / ___ | ___ 3 / ___ ___ | _______________
7 \/ 56 + 14*\/ 2 | 7*\/ 3 \/ 3 *\/ 56 + 14*\/ 2 | 7 \/ 56 + 14*\/ 2 |\/ 3 *\/ 56 + 14*\/ 2 7*\/ 3 | 3 / ___ 14
4 - ------------------ - ------------------ + I*|------------------ - ------------------------| + 4 - ------------------ - ------------------ + I*|------------------------ - ------------------| + 4 + \/ 56 + 14*\/ 2 + ------------------
_______________ 2 | _______________ 2 | _______________ 2 | 2 _______________| _______________
3 / ___ |3 / ___ | 3 / ___ | 3 / ___ | 3 / ___
\/ 56 + 14*\/ 2 \\/ 56 + 14*\/ 2 / \/ 56 + 14*\/ 2 \ \/ 56 + 14*\/ 2 / \/ 56 + 14*\/ 2
$$\left(\frac{14}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + \sqrt[3]{14 \sqrt{2} + 56}\right) + \left(\left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right)\right) + \left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right)\right)\right)$$
=
/ _______________ \ / _______________\
| ___ 3 / ___ ___ | | ___ ___ 3 / ___ |
|\/ 3 *\/ 56 + 14*\/ 2 7*\/ 3 | | 7*\/ 3 \/ 3 *\/ 56 + 14*\/ 2 |
12 + I*|------------------------ - ------------------| + I*|------------------ - ------------------------|
| 2 _______________| | _______________ 2 |
| 3 / ___ | |3 / ___ |
\ \/ 56 + 14*\/ 2 / \\/ 56 + 14*\/ 2 /
$$12 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right) + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right)$$
producto
/ _______________ / _______________\\ / _______________ / _______________ \\ | 3 / ___ | ___ ___ 3 / ___ || | 3 / ___ | ___ 3 / ___ ___ || / _______________ \ | 7 \/ 56 + 14*\/ 2 | 7*\/ 3 \/ 3 *\/ 56 + 14*\/ 2 || | 7 \/ 56 + 14*\/ 2 |\/ 3 *\/ 56 + 14*\/ 2 7*\/ 3 || | 3 / ___ 14 | |4 - ------------------ - ------------------ + I*|------------------ - ------------------------||*|4 - ------------------ - ------------------ + I*|------------------------ - ------------------||*|4 + \/ 56 + 14*\/ 2 + ------------------| | _______________ 2 | _______________ 2 || | _______________ 2 | 2 _______________|| | _______________| | 3 / ___ |3 / ___ || | 3 / ___ | 3 / ___ || | 3 / ___ | \ \/ 56 + 14*\/ 2 \\/ 56 + 14*\/ 2 // \ \/ 56 + 14*\/ 2 \ \/ 56 + 14*\/ 2 // \ \/ 56 + 14*\/ 2 /
$$\left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}} + \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2}\right)\right) \left(- \frac{\sqrt[3]{14 \sqrt{2} + 56}}{2} - \frac{7}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + i \left(- \frac{\sqrt{3} \sqrt[3]{14 \sqrt{2} + 56}}{2} + \frac{7 \sqrt{3}}{\sqrt[3]{14 \sqrt{2} + 56}}\right)\right) \left(\frac{14}{\sqrt[3]{14 \sqrt{2} + 56}} + 4 + \sqrt[3]{14 \sqrt{2} + 56}\right)$$
=
8
$$8$$
8
Respuesta numérica
[src]
x1 = 0.229926352383435 - 0.800228670667707*i
x2 = 0.229926352383435 + 0.800228670667707*i
x3 = 11.5401472952331
x3 = 11.5401472952331