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Ecuación 2*x^2+6*x+9=
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2*x^3-3*x^2-5
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Expresiones idénticas
- dos *x^ tres - tres *x^ dos - cinco = cero
- 2 multiplicar por x al cubo menos 3 multiplicar por x al cuadrado menos 5 es igual a 0
- dos multiplicar por x en el grado tres menos tres multiplicar por x en el grado dos menos cinco es igual a cero
- 2*x3-3*x2-5=0
- 2*x³-3*x²-5=0
- 2*x en el grado 3-3*x en el grado 2-5=0
- 2x^3-3x^2-5=0
- 2x3-3x2-5=0
- 2*x^3-3*x^2-5=O
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Expresiones semejantes
- 2*x^3+3*x^2-5=0
- 2*x^3-3*x^2+5=0
2*x^3-3*x^2-5=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(2 x^{3} - 3 x^{2}\right) - 5 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{3 x^{2}}{2} - \frac{5}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{3}{2}$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - \frac{5}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{3}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - \frac{5}{2}$$
$$\left(2 x^{3} - 3 x^{2}\right) - 5 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{3 x^{2}}{2} - \frac{5}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{3}{2}$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - \frac{5}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{3}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - \frac{5}{2}$$
Suma y producto de raíces
[src]
suma
_____________ / _____________ \ _____________ / _____________ \
/ ____ | / ____ | / ____ | / ____ |
/ 11 \/ 30 | ___ / 11 \/ 30 | / 11 \/ 30 | ___ / 11 \/ 30 | _____________
3 / -- + ------ | \/ 3 *3 / -- + ------ ___ | 3 / -- + ------ |\/ 3 *3 / -- + ------ ___ | / ____
1 \/ 8 4 1 | \/ 8 4 \/ 3 | 1 \/ 8 4 1 | \/ 8 4 \/ 3 | 1 / 11 \/ 30 1
- - ------------------ - -------------------- + I*|- ------------------------ + --------------------| + - - ------------------ - -------------------- + I*|------------------------ - --------------------| + - + 3 / -- + ------ + --------------------
2 2 _____________ | 2 _____________| 2 2 _____________ | 2 _____________| 2 \/ 8 4 _____________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 11 \/ 30 | / 11 \/ 30 | / 11 \/ 30 | / 11 \/ 30 | / 11 \/ 30
8*3 / -- + ------ | 8*3 / -- + ------ | 8*3 / -- + ------ | 8*3 / -- + ------ | 4*3 / -- + ------
\/ 8 4 \ \/ 8 4 / \/ 8 4 \ \/ 8 4 / \/ 8 4
$$\left(\frac{1}{4 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} - \frac{1}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} + \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} - \frac{1}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + i \left(- \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2}\right)\right)\right)$$
=
/ _____________ \ / _____________ \
| / ____ | | / ____ |
| ___ / 11 \/ 30 | | ___ / 11 \/ 30 |
|\/ 3 *3 / -- + ------ ___ | | \/ 3 *3 / -- + ------ ___ |
3 | \/ 8 4 \/ 3 | | \/ 8 4 \/ 3 |
- + I*|------------------------ - --------------------| + I*|- ------------------------ + --------------------|
2 | 2 _____________| | 2 _____________|
| / ____ | | / ____ |
| / 11 \/ 30 | | / 11 \/ 30 |
| 8*3 / -- + ------ | | 8*3 / -- + ------ |
\ \/ 8 4 / \ \/ 8 4 /
$$\frac{3}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} + \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}\right) + i \left(- \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2}\right)$$
producto
/ _____________ / _____________ \\ / _____________ / _____________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 11 \/ 30 | ___ / 11 \/ 30 || | / 11 \/ 30 | ___ / 11 \/ 30 || / _____________ \ | 3 / -- + ------ | \/ 3 *3 / -- + ------ ___ || | 3 / -- + ------ |\/ 3 *3 / -- + ------ ___ || | / ____ | |1 \/ 8 4 1 | \/ 8 4 \/ 3 || |1 \/ 8 4 1 | \/ 8 4 \/ 3 || |1 / 11 \/ 30 1 | |- - ------------------ - -------------------- + I*|- ------------------------ + --------------------||*|- - ------------------ - -------------------- + I*|------------------------ - --------------------||*|- + 3 / -- + ------ + --------------------| |2 2 _____________ | 2 _____________|| |2 2 _____________ | 2 _____________|| |2 \/ 8 4 _____________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 11 \/ 30 | / 11 \/ 30 || | / 11 \/ 30 | / 11 \/ 30 || | / 11 \/ 30 | | 8*3 / -- + ------ | 8*3 / -- + ------ || | 8*3 / -- + ------ | 8*3 / -- + ------ || | 4*3 / -- + ------ | \ \/ 8 4 \ \/ 8 4 // \ \/ 8 4 \ \/ 8 4 // \ \/ 8 4 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} - \frac{1}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + i \left(- \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} - \frac{1}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} + \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}\right)\right) \left(\frac{1}{4 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}\right)$$
=
5/2
$$\frac{5}{2}$$
5/2
Respuesta rápida
[src]
_____________ / _____________ \
/ ____ | / ____ |
/ 11 \/ 30 | ___ / 11 \/ 30 |
3 / -- + ------ | \/ 3 *3 / -- + ------ ___ |
1 \/ 8 4 1 | \/ 8 4 \/ 3 |
x1 = - - ------------------ - -------------------- + I*|- ------------------------ + --------------------|
2 2 _____________ | 2 _____________|
/ ____ | / ____ |
/ 11 \/ 30 | / 11 \/ 30 |
8*3 / -- + ------ | 8*3 / -- + ------ |
\/ 8 4 \ \/ 8 4 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} - \frac{1}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} + \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}\right)$$
_____________ / _____________ \
/ ____ | / ____ |
/ 11 \/ 30 | ___ / 11 \/ 30 |
3 / -- + ------ |\/ 3 *3 / -- + ------ ___ |
1 \/ 8 4 1 | \/ 8 4 \/ 3 |
x2 = - - ------------------ - -------------------- + I*|------------------------ - --------------------|
2 2 _____________ | 2 _____________|
/ ____ | / ____ |
/ 11 \/ 30 | / 11 \/ 30 |
8*3 / -- + ------ | 8*3 / -- + ------ |
\/ 8 4 \ \/ 8 4 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2} - \frac{1}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + i \left(- \frac{\sqrt{3}}{8 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}}{2}\right)$$
_____________
/ ____
1 / 11 \/ 30 1
x3 = - + 3 / -- + ------ + --------------------
2 \/ 8 4 _____________
/ ____
/ 11 \/ 30
4*3 / -- + ------
\/ 8 4
$$x_{3} = \frac{1}{4 \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}} + \frac{1}{2} + \sqrt[3]{\frac{\sqrt{30}}{4} + \frac{11}{8}}$$
x3 = 1/(4*(sqrt(30)/4 + 11/8)^(1/3)) + 1/2 + (sqrt(30)/4 + 11/8)^(1/3)
Respuesta numérica
[src]
x1 = -0.289308444254379 - 1.05783905227298*i
x2 = 2.07861688850876
x3 = -0.289308444254379 + 1.05783905227298*i
x3 = -0.289308444254379 + 1.05783905227298*i
Gráfico