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Ecuación x^2-14*x+33=0
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Ecuación 5(x-4)=3x-10
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- Derivada de:
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x^3-x+3
- Gráfico de la función y =:
- x^3-x+3
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Expresiones idénticas
- x^ tres -x+ tres = cero
- x al cubo menos x más 3 es igual a 0
- x en el grado tres menos x más tres es igual a cero
- x3-x+3=0
- x³-x+3=0
- x en el grado 3-x+3=0
- x^3-x+3=O
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Expresiones semejantes
- x^3+x+3=0
- x^3-x-3=0
x^3-x+3=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 3$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 3$$
Respuesta rápida
[src]
________________ / ________________\
/ _____ | / _____ |
/ 81 3*\/ 717 | ___ / 81 3*\/ 717 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = ----------------------- + --------------------- + I*|- ----------------------- + ---------------------------|
________________ 6 | ________________ 6 |
/ _____ | / _____ |
/ 81 3*\/ 717 | / 81 3*\/ 717 |
2*3 / -- + --------- | 2*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)$$
________________ / ________________\
/ _____ | / _____ |
/ 81 3*\/ 717 | ___ / 81 3*\/ 717 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x2 = ----------------------- + --------------------- + I*|----------------------- - ---------------------------|
________________ 6 | ________________ 6 |
/ _____ | / _____ |
/ 81 3*\/ 717 | / 81 3*\/ 717 |
2*3 / -- + --------- |2*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)$$
________________
/ _____
/ 81 3*\/ 717
3 / -- + ---------
1 \/ 2 2
x3 = - --------------------- - ---------------------
________________ 3
/ _____
/ 81 3*\/ 717
3 / -- + ---------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}$$
x3 = -(3*sqrt(717)/2 + 81/2)^(1/3)/3 - 1/(3*sqrt(717)/2 + 81/2)^(1/3)
Suma y producto de raíces
[src]
suma
________________ / ________________\ ________________ / ________________\ ________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 81 3*\/ 717 | ___ / 81 3*\/ 717 | / 81 3*\/ 717 | ___ / 81 3*\/ 717 | / 81 3*\/ 717
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- | 3 / -- + ---------
1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2
----------------------- + --------------------- + I*|- ----------------------- + ---------------------------| + ----------------------- + --------------------- + I*|----------------------- - ---------------------------| + - --------------------- - ---------------------
________________ 6 | ________________ 6 | ________________ 6 | ________________ 6 | ________________ 3
/ _____ | / _____ | / _____ | / _____ | / _____
/ 81 3*\/ 717 | / 81 3*\/ 717 | / 81 3*\/ 717 | / 81 3*\/ 717 | / 81 3*\/ 717
2*3 / -- + --------- | 2*3 / -- + --------- | 2*3 / -- + --------- |2*3 / -- + --------- | 3 / -- + ---------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right) + \left(\left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)\right) + \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)\right)\right)$$
=
/ ________________\ / ________________\ | / _____ | | / _____ | | ___ / 81 3*\/ 717 | | ___ / 81 3*\/ 717 | | ___ \/ 3 *3 / -- + --------- | | ___ \/ 3 *3 / -- + --------- | | \/ 3 \/ 2 2 | | \/ 3 \/ 2 2 | I*|----------------------- - ---------------------------| + I*|- ----------------------- + ---------------------------| | ________________ 6 | | ________________ 6 | | / _____ | | / _____ | | / 81 3*\/ 717 | | / 81 3*\/ 717 | |2*3 / -- + --------- | | 2*3 / -- + --------- | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________\\ / ________________\ | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 81 3*\/ 717 | ___ / 81 3*\/ 717 || | / 81 3*\/ 717 | ___ / 81 3*\/ 717 || | / 81 3*\/ 717 | | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- || | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- || | 3 / -- + --------- | | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | |----------------------- + --------------------- + I*|- ----------------------- + ---------------------------||*|----------------------- + --------------------- + I*|----------------------- - ---------------------------||*|- --------------------- - ---------------------| | ________________ 6 | ________________ 6 || | ________________ 6 | ________________ 6 || | ________________ 3 | | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 81 3*\/ 717 | / 81 3*\/ 717 || | / 81 3*\/ 717 | / 81 3*\/ 717 || | / 81 3*\/ 717 | |2*3 / -- + --------- | 2*3 / -- + --------- || |2*3 / -- + --------- |2*3 / -- + --------- || | 3 / -- + --------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)\right) \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)$$
=
__________________ ________________ 2/3 __________________ 2/3 2/3 ________________ 2/3
3 / _____ 2/3 3 / _____ 3 ____ / _____\ _____ 3 / _____ 3 ___ / _____\ 3 ___ _____ / _____\ 2/3 _____ 3 / _____ 3 ___ 6 ___ _____ / _____\
81*\/ 324 + 12*\/ 717 81*2 *\/ 81 + 3*\/ 717 3*\/ 18 *\27 + \/ 717 / 3*\/ 717 *\/ 324 + 12*\/ 717 3*\/ 2 *\81 + 3*\/ 717 / \/ 2 *\/ 717 *\81 + 3*\/ 717 / 3*2 *\/ 717 *\/ 81 + 3*\/ 717 \/ 2 *\/ 3 *\/ 239 *\27 + \/ 717 /
-3 + ------------------------ - --------------------------- - -------------------------- - ------------------------------- + --------------------------- - --------------------------------- + ---------------------------------- + -------------------------------------
32 32 8 32 8 72 32 24
$$- \frac{81 \cdot 2^{\frac{2}{3}} \sqrt[3]{3 \sqrt{717} + 81}}{32} - \frac{3 \sqrt{717} \sqrt[3]{12 \sqrt{717} + 324}}{32} - \frac{3 \sqrt[3]{18} \left(\sqrt{717} + 27\right)^{\frac{2}{3}}}{8} - \frac{\sqrt[3]{2} \sqrt{717} \left(3 \sqrt{717} + 81\right)^{\frac{2}{3}}}{72} - 3 + \frac{\sqrt[3]{2} \sqrt{239} \sqrt[6]{3} \left(\sqrt{717} + 27\right)^{\frac{2}{3}}}{24} + \frac{3 \sqrt[3]{2} \left(3 \sqrt{717} + 81\right)^{\frac{2}{3}}}{8} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{717} \sqrt[3]{3 \sqrt{717} + 81}}{32} + \frac{81 \sqrt[3]{12 \sqrt{717} + 324}}{32}$$
-3 + 81*(324 + 12*sqrt(717))^(1/3)/32 - 81*2^(2/3)*(81 + 3*sqrt(717))^(1/3)/32 - 3*18^(1/3)*(27 + sqrt(717))^(2/3)/8 - 3*sqrt(717)*(324 + 12*sqrt(717))^(1/3)/32 + 3*2^(1/3)*(81 + 3*sqrt(717))^(2/3)/8 - 2^(1/3)*sqrt(717)*(81 + 3*sqrt(717))^(2/3)/72 + 3*2^(2/3)*sqrt(717)*(81 + 3*sqrt(717))^(1/3)/32 + 2^(1/3)*3^(1/6)*sqrt(239)*(27 + sqrt(717))^(2/3)/24
Respuesta numérica
[src]
x1 = 0.83584994082858 - 1.04686931884998*i
x2 = -1.67169988165716
x3 = 0.83584994082858 + 1.04686931884998*i
x3 = 0.83584994082858 + 1.04686931884998*i
Gráfico