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- La ecuación:
- Ecuación x+y-3=0
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Ecuación x^2=-3
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Ecuación 3^x=-1
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Ecuación x-11=-7
- Expresar {x} en función de y en la ecuación:
- 4*x+8*y=7
- -12*x+20*y=-3
- -5*x-13*y=-4
- -13*x-7*y=16
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Expresiones idénticas
- x^ tres - seis *x^ dos + catorce *x- siete = cero
- x al cubo menos 6 multiplicar por x al cuadrado más 14 multiplicar por x menos 7 es igual a 0
- x en el grado tres menos seis multiplicar por x en el grado dos más cotangente de angente de orce multiplicar por x menos siete es igual a cero
- x3-6*x2+14*x-7=0
- x³-6*x²+14*x-7=0
- x en el grado 3-6*x en el grado 2+14*x-7=0
- x^3-6x^2+14x-7=0
- x3-6x2+14x-7=0
- x^3-6*x^2+14*x-7=O
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Expresiones semejantes
- x^3-6*x^2+14*x+7=0
- x^3-6*x^2-14*x-7=0
- x^3+6*x^2+14*x-7=0
x^3-6*x^2+14*x-7=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - 6*x + 14*x - 7 = 0
$$\left(14 x + \left(x^{3} - 6 x^{2}\right)\right) - 7 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 14$$
$$v = \frac{d}{a}$$
$$v = -7$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 6$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 14$$
$$x_{1} x_{2} x_{3} = -7$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 14$$
$$v = \frac{d}{a}$$
$$v = -7$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 6$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 14$$
$$x_{1} x_{2} x_{3} = -7$$
Suma y producto de raíces
[src]
suma
__________________ / __________________\ __________________ / __________________\ __________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 135 3*\/ 2121 | ___ / 135 3*\/ 2121 | / 135 3*\/ 2121 | ___ / 135 3*\/ 2121 | / 135 3*\/ 2121
3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- | 3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- | 3 / --- + ----------
1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2 | \/ 3 \/ 2 2 | 2 \/ 2 2
2 - ----------------------- + ----------------------- + I*|----------------------- + -----------------------------| + 2 - ----------------------- + ----------------------- + I*|- ----------------------- - -----------------------------| + 2 + ----------------------- - -----------------------
__________________ 6 | __________________ 6 | __________________ 6 | __________________ 6 | __________________ 3
/ ______ | / ______ | / ______ | / ______ | / ______
/ 135 3*\/ 2121 | / 135 3*\/ 2121 | / 135 3*\/ 2121 | / 135 3*\/ 2121 | / 135 3*\/ 2121
3 / --- + ---------- |3 / --- + ---------- | 3 / --- + ---------- | 3 / --- + ---------- | 3 / --- + ----------
\/ 2 2 \\/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{3} + \frac{2}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + 2\right) + \left(\left(- \frac{1}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} + 2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}\right)\right) + \left(- \frac{1}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} + 2 + i \left(\frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6}\right)\right)\right)$$
=
/ __________________\ / __________________\
| / ______ | | / ______ |
| ___ / 135 3*\/ 2121 | | ___ / 135 3*\/ 2121 |
| ___ \/ 3 *3 / --- + ---------- | | ___ \/ 3 *3 / --- + ---------- |
| \/ 3 \/ 2 2 | | \/ 3 \/ 2 2 |
6 + I*|----------------------- + -----------------------------| + I*|- ----------------------- - -----------------------------|
| __________________ 6 | | __________________ 6 |
| / ______ | | / ______ |
| / 135 3*\/ 2121 | | / 135 3*\/ 2121 |
|3 / --- + ---------- | | 3 / --- + ---------- |
\\/ 2 2 / \ \/ 2 2 /
$$6 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}\right) + i \left(\frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6}\right)$$
producto
/ __________________ / __________________\\ / __________________ / __________________\\ / __________________\ | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 135 3*\/ 2121 | ___ / 135 3*\/ 2121 || | / 135 3*\/ 2121 | ___ / 135 3*\/ 2121 || | / 135 3*\/ 2121 | | 3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- || | 3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- || | 3 / --- + ---------- | | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 2 \/ 2 2 | |2 - ----------------------- + ----------------------- + I*|----------------------- + -----------------------------||*|2 - ----------------------- + ----------------------- + I*|- ----------------------- - -----------------------------||*|2 + ----------------------- - -----------------------| | __________________ 6 | __________________ 6 || | __________________ 6 | __________________ 6 || | __________________ 3 | | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 135 3*\/ 2121 | / 135 3*\/ 2121 || | / 135 3*\/ 2121 | / 135 3*\/ 2121 || | / 135 3*\/ 2121 | | 3 / --- + ---------- |3 / --- + ---------- || | 3 / --- + ---------- | 3 / --- + ---------- || | 3 / --- + ---------- | \ \/ 2 2 \\/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(- \frac{1}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} + 2 + i \left(\frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6}\right)\right) \left(- \frac{1}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} + 2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{3} + \frac{2}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + 2\right)$$
=
___________________ __________________ ___________________ 2/3 2/3 2/3 __________________ 2/3
3 / ______ 2/3 3 / ______ ______ 3 / ______ 3 ___ / ______\ 3 ____ / ______\ 3 ___ ______ / ______\ 2/3 ______ 3 / ______ 3 ___ 6 ___ _____ / ______\
245*\/ 540 + 12*\/ 2121 245*2 *\/ 135 + 3*\/ 2121 49*\/ 2121 *\/ 540 + 12*\/ 2121 35*\/ 2 *\135 + 3*\/ 2121 / 35*\/ 18 *\45 + \/ 2121 / 7*\/ 2 *\/ 2121 *\135 + 3*\/ 2121 / 49*2 *\/ 2121 *\/ 135 + 3*\/ 2121 7*\/ 2 *\/ 3 *\/ 707 *\45 + \/ 2121 /
7 + -------------------------- - ------------------------------ - ---------------------------------- - ------------------------------ + ---------------------------- + -------------------------------------- + -------------------------------------- - ----------------------------------------
16 16 144 24 24 216 144 72
$$- \frac{49 \sqrt{2121} \sqrt[3]{540 + 12 \sqrt{2121}}}{144} - \frac{245 \cdot 2^{\frac{2}{3}} \sqrt[3]{135 + 3 \sqrt{2121}}}{16} - \frac{7 \sqrt[3]{2} \sqrt[6]{3} \sqrt{707} \left(45 + \sqrt{2121}\right)^{\frac{2}{3}}}{72} - \frac{35 \sqrt[3]{2} \left(135 + 3 \sqrt{2121}\right)^{\frac{2}{3}}}{24} + 7 + \frac{35 \sqrt[3]{18} \left(45 + \sqrt{2121}\right)^{\frac{2}{3}}}{24} + \frac{7 \sqrt[3]{2} \sqrt{2121} \left(135 + 3 \sqrt{2121}\right)^{\frac{2}{3}}}{216} + \frac{245 \sqrt[3]{540 + 12 \sqrt{2121}}}{16} + \frac{49 \cdot 2^{\frac{2}{3}} \sqrt{2121} \sqrt[3]{135 + 3 \sqrt{2121}}}{144}$$
7 + 245*(540 + 12*sqrt(2121))^(1/3)/16 - 245*2^(2/3)*(135 + 3*sqrt(2121))^(1/3)/16 - 49*sqrt(2121)*(540 + 12*sqrt(2121))^(1/3)/144 - 35*2^(1/3)*(135 + 3*sqrt(2121))^(2/3)/24 + 35*18^(1/3)*(45 + sqrt(2121))^(2/3)/24 + 7*2^(1/3)*sqrt(2121)*(135 + 3*sqrt(2121))^(2/3)/216 + 49*2^(2/3)*sqrt(2121)*(135 + 3*sqrt(2121))^(1/3)/144 - 7*2^(1/3)*3^(1/6)*sqrt(707)*(45 + sqrt(2121))^(2/3)/72
Respuesta rápida
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__________________ / __________________\
/ ______ | / ______ |
/ 135 3*\/ 2121 | ___ / 135 3*\/ 2121 |
3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = 2 - ----------------------- + ----------------------- + I*|----------------------- + -----------------------------|
__________________ 6 | __________________ 6 |
/ ______ | / ______ |
/ 135 3*\/ 2121 | / 135 3*\/ 2121 |
3 / --- + ---------- |3 / --- + ---------- |
\/ 2 2 \\/ 2 2 /
$$x_{1} = - \frac{1}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} + 2 + i \left(\frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6}\right)$$
__________________ / __________________\
/ ______ | / ______ |
/ 135 3*\/ 2121 | ___ / 135 3*\/ 2121 |
3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x2 = 2 - ----------------------- + ----------------------- + I*|- ----------------------- - -----------------------------|
__________________ 6 | __________________ 6 |
/ ______ | / ______ |
/ 135 3*\/ 2121 | / 135 3*\/ 2121 |
3 / --- + ---------- | 3 / --- + ---------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{1}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} + 2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{6} - \frac{\sqrt{3}}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}\right)$$
__________________
/ ______
/ 135 3*\/ 2121
3 / --- + ----------
2 \/ 2 2
x3 = 2 + ----------------------- - -----------------------
__________________ 3
/ ______
/ 135 3*\/ 2121
3 / --- + ----------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}}{3} + \frac{2}{\sqrt[3]{\frac{135}{2} + \frac{3 \sqrt{2121}}{2}}} + 2$$
x3 = -(135/2 + 3*sqrt(2121)/2)^(1/3)/3 + 2/(135/2 + 3*sqrt(2121)/2)^(1/3) + 2
Respuesta numérica
[src]
x1 = 0.671731144331392
x2 = 2.6641344278343 - 1.82297109541111*i
x3 = 2.6641344278343 + 1.82297109541111*i
x3 = 2.6641344278343 + 1.82297109541111*i
Gráfico