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Ecuación 6*x^2+x-7=0
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- Integral de d{x}:
- x^3+x+3
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Expresiones idénticas
- x^ tres +x+ tres = cero
- x al cubo más x más 3 es igual a 0
- x en el grado tres más 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
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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
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = 3$$
Suma y producto de raíces
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suma
________________ / ________________\ ________________ / ________________\ ________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 81 3*\/ 741 | ___ / 81 3*\/ 741 | / 81 3*\/ 741 | ___ / 81 3*\/ 741 | / 81 3*\/ 741
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*\/ 741 | / 81 3*\/ 741 | / 81 3*\/ 741 | / 81 3*\/ 741 | / 81 3*\/ 741
2*3 / -- + --------- |2*3 / -- + --------- | 2*3 / -- + --------- | 2*3 / -- + --------- | 3 / -- + ---------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right) + \left(\left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right)\right) + \left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right)\right)\right)$$
=
/ ________________\ / ________________\ | / _____ | | / _____ | | ___ / 81 3*\/ 741 | | ___ / 81 3*\/ 741 | | ___ \/ 3 *3 / -- + --------- | | ___ \/ 3 *3 / -- + --------- | | \/ 3 \/ 2 2 | | \/ 3 \/ 2 2 | I*|----------------------- + ---------------------------| + I*|- ----------------------- - ---------------------------| | ________________ 6 | | ________________ 6 | | / _____ | | / _____ | | / 81 3*\/ 741 | | / 81 3*\/ 741 | |2*3 / -- + --------- | | 2*3 / -- + --------- | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________\\ / ________________\ | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 81 3*\/ 741 | ___ / 81 3*\/ 741 || | / 81 3*\/ 741 | ___ / 81 3*\/ 741 || | / 81 3*\/ 741 | | 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*\/ 741 | / 81 3*\/ 741 || | / 81 3*\/ 741 | / 81 3*\/ 741 || | / 81 3*\/ 741 | | 2*3 / -- + --------- |2*3 / -- + --------- || | 2*3 / -- + --------- | 2*3 / -- + --------- || |3 / -- + --------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \\/ 2 2 /
$$\left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right)\right) \left(- \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right)$$
=
__________________ ________________ 2/3 __________________ 2/3 2/3 ________________ 2/3
3 / _____ 2/3 3 / _____ 3 ___ / _____\ _____ 3 / _____ 3 ____ / _____\ 3 ___ _____ / _____\ 2/3 _____ 3 / _____ 3 ___ 6 ___ _____ / _____\
81*\/ 324 + 12*\/ 741 81*2 *\/ 81 + 3*\/ 741 3*\/ 2 *\81 + 3*\/ 741 / 3*\/ 741 *\/ 324 + 12*\/ 741 3*\/ 18 *\27 + \/ 741 / \/ 2 *\/ 741 *\81 + 3*\/ 741 / 3*2 *\/ 741 *\/ 81 + 3*\/ 741 \/ 2 *\/ 3 *\/ 247 *\27 + \/ 741 /
-3 + ------------------------ - --------------------------- - --------------------------- - ------------------------------- + -------------------------- + --------------------------------- + ---------------------------------- - -------------------------------------
32 32 8 32 8 72 32 24
$$- \frac{3 \sqrt{741} \sqrt[3]{324 + 12 \sqrt{741}}}{32} - \frac{81 \cdot 2^{\frac{2}{3}} \sqrt[3]{81 + 3 \sqrt{741}}}{32} - \frac{\sqrt[3]{2} \sqrt{247} \sqrt[6]{3} \left(27 + \sqrt{741}\right)^{\frac{2}{3}}}{24} - \frac{3 \sqrt[3]{2} \left(81 + 3 \sqrt{741}\right)^{\frac{2}{3}}}{8} - 3 + \frac{3 \sqrt[3]{18} \left(27 + \sqrt{741}\right)^{\frac{2}{3}}}{8} + \frac{\sqrt[3]{2} \sqrt{741} \left(81 + 3 \sqrt{741}\right)^{\frac{2}{3}}}{72} + \frac{81 \sqrt[3]{324 + 12 \sqrt{741}}}{32} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{741} \sqrt[3]{81 + 3 \sqrt{741}}}{32}$$
-3 + 81*(324 + 12*sqrt(741))^(1/3)/32 - 81*2^(2/3)*(81 + 3*sqrt(741))^(1/3)/32 - 3*2^(1/3)*(81 + 3*sqrt(741))^(2/3)/8 - 3*sqrt(741)*(324 + 12*sqrt(741))^(1/3)/32 + 3*18^(1/3)*(27 + sqrt(741))^(2/3)/8 + 2^(1/3)*sqrt(741)*(81 + 3*sqrt(741))^(2/3)/72 + 3*2^(2/3)*sqrt(741)*(81 + 3*sqrt(741))^(1/3)/32 - 2^(1/3)*3^(1/6)*sqrt(247)*(27 + sqrt(741))^(2/3)/24
Respuesta rápida
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________________ / ________________\
/ _____ | / _____ |
/ 81 3*\/ 741 | ___ / 81 3*\/ 741 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = - ----------------------- + --------------------- + I*|----------------------- + ---------------------------|
________________ 6 | ________________ 6 |
/ _____ | / _____ |
/ 81 3*\/ 741 | / 81 3*\/ 741 |
2*3 / -- + --------- |2*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6}\right)$$
________________ / ________________\
/ _____ | / _____ |
/ 81 3*\/ 741 | ___ / 81 3*\/ 741 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x2 = - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------|
________________ 6 | ________________ 6 |
/ _____ | / _____ |
/ 81 3*\/ 741 | / 81 3*\/ 741 |
2*3 / -- + --------- | 2*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{1}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}} + \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}\right)$$
________________
/ _____
/ 81 3*\/ 741
3 / -- + ---------
1 \/ 2 2
x3 = --------------------- - ---------------------
________________ 3
/ _____
/ 81 3*\/ 741
3 / -- + ---------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{81}{2} + \frac{3 \sqrt{741}}{2}}}$$
x3 = -(81/2 + 3*sqrt(741)/2)^(1/3)/3 + (81/2 + 3*sqrt(741)/2)^(-1/3)
Respuesta numérica
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x1 = 0.606705831381115 + 1.45061224918844*i
x2 = 0.606705831381115 - 1.45061224918844*i
x3 = -1.21341166276223
x3 = -1.21341166276223