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- La ecuación:
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Ecuación 5x+15=x-1
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Ecuación 3x^2=0
- Ecuación x^2-49=0
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Ecuación 1-x/2=x/3
- Expresar {x} en función de y en la ecuación:
- 4*x-1*y=-3
- -11*x+9*y=-20
- 9*x+17*y=13
- -19*x+14*y=20
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Expresiones idénticas
- (5x^ tres + dos x^ dos +3x- uno)/(2)
- (5x al cubo más 2x al cuadrado más 3x menos 1) dividir por (2)
- (5x en el grado tres más dos x en el grado dos más 3x menos uno) dividir por (2)
- (5x3+2x2+3x-1)/(2)
- 5x3+2x2+3x-1/2
- (5x³+2x²+3x-1)/(2)
- (5x en el grado 3+2x en el grado 2+3x-1)/(2)
- 5x^3+2x^2+3x-1/2
- (5x^3+2x^2+3x-1) dividir por (2)
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Expresiones semejantes
- (5x^3-2x^2+3x-1)/(2)
- (5x^3+2x^2+3x+1)/(2)
- (5x^3+2x^2-3x-1)/(2)
(5x^3+2x^2+3x-1)/(2) la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
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3 2
5*x + 2*x + 3*x - 1
--------------------- = 0
2
$$\frac{\left(3 x + \left(5 x^{3} + 2 x^{2}\right)\right) - 1}{2} = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\frac{\left(3 x + \left(5 x^{3} + 2 x^{2}\right)\right) - 1}{2} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{2 x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2}{5}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{5}$$
$$v = \frac{d}{a}$$
$$v = - \frac{1}{5}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{2}{5}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{3}{5}$$
$$x_{1} x_{2} x_{3} = - \frac{1}{5}$$
$$\frac{\left(3 x + \left(5 x^{3} + 2 x^{2}\right)\right) - 1}{2} = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{2 x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{2}{5}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{5}$$
$$v = \frac{d}{a}$$
$$v = - \frac{1}{5}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{2}{5}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{3}{5}$$
$$x_{1} x_{2} x_{3} = - \frac{1}{5}$$
Suma y producto de raíces
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suma
_________________ / _________________\ _________________ / _________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 929 \/ 5061 | ___ / 929 \/ 5061 | / 929 \/ 5061 | ___ / 929 \/ 5061 | _________________
3 / ---- + -------- | ___ \/ 3 *3 / ---- + -------- | 3 / ---- + -------- |\/ 3 *3 / ---- + -------- ___ | / ______
2 \/ 6750 450 41 | 41*\/ 3 \/ 6750 450 | 2 \/ 6750 450 41 | \/ 6750 450 41*\/ 3 | 2 / 929 \/ 5061 41
- -- - ---------------------- + -------------------------- + I*|- -------------------------- - ----------------------------| + - -- - ---------------------- + -------------------------- + I*|---------------------------- + --------------------------| + - -- + 3 / ---- + -------- - --------------------------
15 2 _________________ | _________________ 2 | 15 2 _________________ | 2 _________________| 15 \/ 6750 450 _________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 929 \/ 5061 | / 929 \/ 5061 | / 929 \/ 5061 | / 929 \/ 5061 | / 929 \/ 5061
450*3 / ---- + -------- | 450*3 / ---- + -------- | 450*3 / ---- + -------- | 450*3 / ---- + -------- | 225*3 / ---- + --------
\/ 6750 450 \ \/ 6750 450 / \/ 6750 450 \ \/ 6750 450 / \/ 6750 450
$$\left(- \frac{41}{225 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} - \frac{2}{15} + \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{2}{15} + \frac{41}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{2}{15} + \frac{41}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + i \left(\frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2}\right)\right)\right)$$
=
/ _________________ \ / _________________\
| / ______ | | / ______ |
| ___ / 929 \/ 5061 | | ___ / 929 \/ 5061 |
|\/ 3 *3 / ---- + -------- ___ | | ___ \/ 3 *3 / ---- + -------- |
2 | \/ 6750 450 41*\/ 3 | | 41*\/ 3 \/ 6750 450 |
- - + I*|---------------------------- + --------------------------| + I*|- -------------------------- - ----------------------------|
5 | 2 _________________| | _________________ 2 |
| / ______ | | / ______ |
| / 929 \/ 5061 | | / 929 \/ 5061 |
| 450*3 / ---- + -------- | | 450*3 / ---- + -------- |
\ \/ 6750 450 / \ \/ 6750 450 /
$$- \frac{2}{5} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}\right) + i \left(\frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2}\right)$$
producto
/ _________________ / _________________\\ / _________________ / _________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 929 \/ 5061 | ___ / 929 \/ 5061 || | / 929 \/ 5061 | ___ / 929 \/ 5061 || / _________________ \ | 3 / ---- + -------- | ___ \/ 3 *3 / ---- + -------- || | 3 / ---- + -------- |\/ 3 *3 / ---- + -------- ___ || | / ______ | | 2 \/ 6750 450 41 | 41*\/ 3 \/ 6750 450 || | 2 \/ 6750 450 41 | \/ 6750 450 41*\/ 3 || | 2 / 929 \/ 5061 41 | |- -- - ---------------------- + -------------------------- + I*|- -------------------------- - ----------------------------||*|- -- - ---------------------- + -------------------------- + I*|---------------------------- + --------------------------||*|- -- + 3 / ---- + -------- - --------------------------| | 15 2 _________________ | _________________ 2 || | 15 2 _________________ | 2 _________________|| | 15 \/ 6750 450 _________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 929 \/ 5061 | / 929 \/ 5061 || | / 929 \/ 5061 | / 929 \/ 5061 || | / 929 \/ 5061 | | 450*3 / ---- + -------- | 450*3 / ---- + -------- || | 450*3 / ---- + -------- | 450*3 / ---- + -------- || | 225*3 / ---- + -------- | \ \/ 6750 450 \ \/ 6750 450 // \ \/ 6750 450 \ \/ 6750 450 // \ \/ 6750 450 /
$$\left(- \frac{\sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{2}{15} + \frac{41}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + i \left(\frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{2}{15} + \frac{41}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}\right)\right) \left(- \frac{41}{225 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} - \frac{2}{15} + \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}\right)$$
=
1/5
$$\frac{1}{5}$$
1/5
Respuesta rápida
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_________________ / _________________\
/ ______ | / ______ |
/ 929 \/ 5061 | ___ / 929 \/ 5061 |
3 / ---- + -------- | ___ \/ 3 *3 / ---- + -------- |
2 \/ 6750 450 41 | 41*\/ 3 \/ 6750 450 |
x1 = - -- - ---------------------- + -------------------------- + I*|- -------------------------- - ----------------------------|
15 2 _________________ | _________________ 2 |
/ ______ | / ______ |
/ 929 \/ 5061 | / 929 \/ 5061 |
450*3 / ---- + -------- | 450*3 / ---- + -------- |
\/ 6750 450 \ \/ 6750 450 /
$$x_{1} = - \frac{\sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{2}{15} + \frac{41}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}\right)$$
_________________ / _________________ \
/ ______ | / ______ |
/ 929 \/ 5061 | ___ / 929 \/ 5061 |
3 / ---- + -------- |\/ 3 *3 / ---- + -------- ___ |
2 \/ 6750 450 41 | \/ 6750 450 41*\/ 3 |
x2 = - -- - ---------------------- + -------------------------- + I*|---------------------------- + --------------------------|
15 2 _________________ | 2 _________________|
/ ______ | / ______ |
/ 929 \/ 5061 | / 929 \/ 5061 |
450*3 / ---- + -------- | 450*3 / ---- + -------- |
\/ 6750 450 \ \/ 6750 450 /
$$x_{2} = - \frac{\sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2} - \frac{2}{15} + \frac{41}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + i \left(\frac{41 \sqrt{3}}{450 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} + \frac{\sqrt{3} \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}}{2}\right)$$
_________________
/ ______
2 / 929 \/ 5061 41
x3 = - -- + 3 / ---- + -------- - --------------------------
15 \/ 6750 450 _________________
/ ______
/ 929 \/ 5061
225*3 / ---- + --------
\/ 6750 450
$$x_{3} = - \frac{41}{225 \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}} - \frac{2}{15} + \sqrt[3]{\frac{929}{6750} + \frac{\sqrt{5061}}{450}}$$
x3 = -41/(225*(929/6750 + sqrt(5061)/450)^(1/3)) - 2/15 + (929/6750 + sqrt(5061)/450)^(1/3)
Respuesta numérica
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x1 = -0.329695073882822 - 0.813843023021972*i
x2 = 0.259390147765645
x3 = -0.329695073882822 + 0.813843023021972*i
x3 = -0.329695073882822 + 0.813843023021972*i