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Ecuación x^2+5x=36
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Ecuación 3*x=8
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Ecuación (x-87)-27=36
- Ecuación (x^2-16)^2+(x^2+x-20)^2=0
- Expresar {x} en función de y en la ecuación:
- -17*x+2*y=9
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Expresiones idénticas
- seis x^ tres +3sqrt(dos)*x^ dos -6*x+ dos sqrt(2)= cero
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- seis x en el grado tres más 3 raíz cuadrada de (dos) multiplicar por x en el grado dos menos 6 multiplicar por x más dos raíz cuadrada de (2) es igual a cero
- 6x^3+3√(2)*x^2-6*x+2√(2)=0
- 6x3+3sqrt(2)*x2-6*x+2sqrt(2)=0
- 6x3+3sqrt2*x2-6*x+2sqrt2=0
- 6x³+3sqrt(2)*x²-6*x+2sqrt(2)=0
- 6x en el grado 3+3sqrt(2)*x en el grado 2-6*x+2sqrt(2)=0
- 6x^3+3sqrt(2)x^2-6x+2sqrt(2)=0
- 6x3+3sqrt(2)x2-6x+2sqrt(2)=0
- 6x3+3sqrt2x2-6x+2sqrt2=0
- 6x^3+3sqrt2x^2-6x+2sqrt2=0
- 6x^3+3sqrt(2)*x^2-6*x+2sqrt(2)=O
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Expresiones semejantes
- 6x^3-3sqrt(2)*x^2-6*x+2sqrt(2)=0
- 6x^3+3sqrt(2)*x^2+6*x+2sqrt(2)=0
- 6x^3+3sqrt(2)*x^2-6*x-2sqrt(2)=0
6x^3+3sqrt(2)*x^2-6*x+2sqrt(2)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 ___ 2 ___ 6*x + 3*\/ 2 *x - 6*x + 2*\/ 2 = 0
$$\left(- 6 x + \left(6 x^{3} + 3 \sqrt{2} x^{2}\right)\right) + 2 \sqrt{2} = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 6 x + \left(6 x^{3} + 3 \sqrt{2} x^{2}\right)\right) + 2 \sqrt{2} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{\sqrt{2} x^{2}}{2} - x + \frac{\sqrt{2}}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{\sqrt{2}}{2}$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = \frac{\sqrt{2}}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{\sqrt{2}}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = \frac{\sqrt{2}}{3}$$
$$\left(- 6 x + \left(6 x^{3} + 3 \sqrt{2} x^{2}\right)\right) + 2 \sqrt{2} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{\sqrt{2} x^{2}}{2} - x + \frac{\sqrt{2}}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{\sqrt{2}}{2}$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = \frac{\sqrt{2}}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{\sqrt{2}}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = \frac{\sqrt{2}}{3}$$
Respuesta rápida
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___ / ___ 3 ___ ___ 2/3\ ___ 3 ___ ___ 2/3
\/ 2 | \/ 6 *\/ 7 \/ 6 *7 | \/ 2 *\/ 7 \/ 2 *7
x1 = - ----- + I*|- ----------- + ----------| + ----------- + ----------
6 \ 12 12 / 12 12
$$x_{1} = - \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right)$$
___ / ___ 2/3 ___ 3 ___\ ___ 3 ___ ___ 2/3
\/ 2 | \/ 6 *7 \/ 6 *\/ 7 | \/ 2 *\/ 7 \/ 2 *7
x2 = - ----- + I*|- ---------- + -----------| + ----------- + ----------
6 \ 12 12 / 12 12
$$x_{2} = - \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right)$$
___ ___ 3 ___ ___ 2/3
\/ 2 \/ 2 *\/ 7 \/ 2 *4*7
x3 = - ----- - ----------- - ------------
6 6 24
$$x_{3} = - \frac{\sqrt{2} \cdot 4 \cdot 7^{\frac{2}{3}}}{24} - \frac{\sqrt{2} \sqrt[3]{7}}{6} - \frac{\sqrt{2}}{6}$$
x3 = -sqrt(2)*4*7^(2/3)/24 - sqrt(2)*7^(1/3)/6 - sqrt(2)/6
Suma y producto de raíces
[src]
suma
___ / ___ 3 ___ ___ 2/3\ ___ 3 ___ ___ 2/3 ___ / ___ 2/3 ___ 3 ___\ ___ 3 ___ ___ 2/3 ___ ___ 3 ___ ___ 2/3
\/ 2 | \/ 6 *\/ 7 \/ 6 *7 | \/ 2 *\/ 7 \/ 2 *7 \/ 2 | \/ 6 *7 \/ 6 *\/ 7 | \/ 2 *\/ 7 \/ 2 *7 \/ 2 \/ 2 *\/ 7 \/ 2 *4*7
- ----- + I*|- ----------- + ----------| + ----------- + ---------- + - ----- + I*|- ---------- + -----------| + ----------- + ---------- + - ----- - ----------- - ------------
6 \ 12 12 / 12 12 6 \ 12 12 / 12 12 6 6 24
$$\left(- \frac{\sqrt{2} \cdot 4 \cdot 7^{\frac{2}{3}}}{24} - \frac{\sqrt{2} \sqrt[3]{7}}{6} - \frac{\sqrt{2}}{6}\right) + \left(\left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right)\right) + \left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right)\right)\right)$$
=
___ / ___ 3 ___ ___ 2/3\ / ___ 2/3 ___ 3 ___\
\/ 2 | \/ 6 *\/ 7 \/ 6 *7 | | \/ 6 *7 \/ 6 *\/ 7 |
- ----- + I*|- ----------- + ----------| + I*|- ---------- + -----------|
2 \ 12 12 / \ 12 12 /
$$- \frac{\sqrt{2}}{2} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right) + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right)$$
producto
/ ___ / ___ 3 ___ ___ 2/3\ ___ 3 ___ ___ 2/3\ / ___ / ___ 2/3 ___ 3 ___\ ___ 3 ___ ___ 2/3\ / ___ ___ 3 ___ ___ 2/3\ | \/ 2 | \/ 6 *\/ 7 \/ 6 *7 | \/ 2 *\/ 7 \/ 2 *7 | | \/ 2 | \/ 6 *7 \/ 6 *\/ 7 | \/ 2 *\/ 7 \/ 2 *7 | | \/ 2 \/ 2 *\/ 7 \/ 2 *4*7 | |- ----- + I*|- ----------- + ----------| + ----------- + ----------|*|- ----- + I*|- ---------- + -----------| + ----------- + ----------|*|- ----- - ----------- - ------------| \ 6 \ 12 12 / 12 12 / \ 6 \ 12 12 / 12 12 / \ 6 6 24 /
$$\left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \sqrt[3]{7}}{12} + \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12}\right)\right) \left(- \frac{\sqrt{2}}{6} + \frac{\sqrt{2} \sqrt[3]{7}}{12} + \frac{\sqrt{2} \cdot 7^{\frac{2}{3}}}{12} + i \left(- \frac{\sqrt{6} \cdot 7^{\frac{2}{3}}}{12} + \frac{\sqrt{6} \sqrt[3]{7}}{12}\right)\right) \left(- \frac{\sqrt{2} \cdot 4 \cdot 7^{\frac{2}{3}}}{24} - \frac{\sqrt{2} \sqrt[3]{7}}{6} - \frac{\sqrt{2}}{6}\right)$$
=
___ -\/ 2 ------- 3
$$- \frac{\sqrt{2}}{3}$$
-sqrt(2)/3
Respuesta numérica
[src]
x1 = 0.42099215515006 + 0.356477207629844*i
x2 = 0.42099215515006 - 0.356477207629844*i
x3 = -1.54909109148667
x3 = -1.54909109148667