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Ecuación x^2-6x+9=0
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Ecuación (x-11)^4=(x+3)^4
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Ecuación -x/12+x=55/12
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Expresiones idénticas
- x^ tres -5x^ dos -9x- cuarenta y cinco = cero
- x al cubo menos 5x al cuadrado menos 9x menos 45 es igual a 0
- x en el grado tres menos 5x en el grado dos menos 9x menos cuarenta y cinco es igual a cero
- x3-5x2-9x-45=0
- x³-5x²-9x-45=0
- x en el grado 3-5x en el grado 2-9x-45=0
- x^3-5x^2-9x-45=O
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Expresiones semejantes
- x^3-5x^2-9x+45=0
- x^3+5x^2-9x-45=0
- x^3-5x^2+9x-45=0
x^3-5x^2-9x-45=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - 5*x - 9*x - 45 = 0
$$\left(- 9 x + \left(x^{3} - 5 x^{2}\right)\right) - 45 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -45$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -45$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -45$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -45$$
Respuesta rápida
[src]
________________ / ________________ \
/ ______ | / ______ |
/ 935 \/ 9057 | ___ / 935 \/ 9057 |
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ |
5 26 \/ 27 3 | \/ 27 3 26*\/ 3 |
x1 = - - ----------------------- - --------------------- + I*|- --------------------------- + -----------------------|
3 ________________ 2 | 2 ________________|
/ ______ | / ______ |
/ 935 \/ 9057 | / 935 \/ 9057 |
9*3 / --- + -------- | 9*3 / --- + -------- |
\/ 27 3 \ \/ 27 3 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} - \frac{26}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} + \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}\right)$$
________________ / ________________ \
/ ______ | / ______ |
/ 935 \/ 9057 | ___ / 935 \/ 9057 |
3 / --- + -------- |\/ 3 *3 / --- + -------- ___ |
5 26 \/ 27 3 | \/ 27 3 26*\/ 3 |
x2 = - - ----------------------- - --------------------- + I*|--------------------------- - -----------------------|
3 ________________ 2 | 2 ________________|
/ ______ | / ______ |
/ 935 \/ 9057 | / 935 \/ 9057 |
9*3 / --- + -------- | 9*3 / --- + -------- |
\/ 27 3 \ \/ 27 3 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} - \frac{26}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + i \left(- \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2}\right)$$
________________
/ ______
5 / 935 \/ 9057 52
x3 = - + 3 / --- + -------- + -----------------------
3 \/ 27 3 ________________
/ ______
/ 935 \/ 9057
9*3 / --- + --------
\/ 27 3
$$x_{3} = \frac{52}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}$$
x3 = 52/(9*(sqrt(9057)/3 + 935/27)^(1/3)) + 5/3 + (sqrt(9057)/3 + 935/27)^(1/3)
Suma y producto de raíces
[src]
suma
________________ / ________________ \ ________________ / ________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 935 \/ 9057 | ___ / 935 \/ 9057 | / 935 \/ 9057 | ___ / 935 \/ 9057 | ________________
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ | / ______
5 26 \/ 27 3 | \/ 27 3 26*\/ 3 | 5 26 \/ 27 3 | \/ 27 3 26*\/ 3 | 5 / 935 \/ 9057 52
- - ----------------------- - --------------------- + I*|- --------------------------- + -----------------------| + - - ----------------------- - --------------------- + I*|--------------------------- - -----------------------| + - + 3 / --- + -------- + -----------------------
3 ________________ 2 | 2 ________________| 3 ________________ 2 | 2 ________________| 3 \/ 27 3 ________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 935 \/ 9057 | / 935 \/ 9057 | / 935 \/ 9057 | / 935 \/ 9057 | / 935 \/ 9057
9*3 / --- + -------- | 9*3 / --- + -------- | 9*3 / --- + -------- | 9*3 / --- + -------- | 9*3 / --- + --------
\/ 27 3 \ \/ 27 3 / \/ 27 3 \ \/ 27 3 / \/ 27 3
$$\left(\frac{52}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} - \frac{26}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} + \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} - \frac{26}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + i \left(- \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________ \
| / ______ | | / ______ |
| ___ / 935 \/ 9057 | | ___ / 935 \/ 9057 |
|\/ 3 *3 / --- + -------- ___ | | \/ 3 *3 / --- + -------- ___ |
| \/ 27 3 26*\/ 3 | | \/ 27 3 26*\/ 3 |
5 + I*|--------------------------- - -----------------------| + I*|- --------------------------- + -----------------------|
| 2 ________________| | 2 ________________|
| / ______ | | / ______ |
| / 935 \/ 9057 | | / 935 \/ 9057 |
| 9*3 / --- + -------- | | 9*3 / --- + -------- |
\ \/ 27 3 / \ \/ 27 3 /
$$5 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} + \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}\right) + i \left(- \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2}\right)$$
producto
/ ________________ / ________________ \\ / ________________ / ________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 935 \/ 9057 | ___ / 935 \/ 9057 || | / 935 \/ 9057 | ___ / 935 \/ 9057 || / ________________ \ | 3 / --- + -------- | \/ 3 *3 / --- + -------- ___ || | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ || | / ______ | |5 26 \/ 27 3 | \/ 27 3 26*\/ 3 || |5 26 \/ 27 3 | \/ 27 3 26*\/ 3 || |5 / 935 \/ 9057 52 | |- - ----------------------- - --------------------- + I*|- --------------------------- + -----------------------||*|- - ----------------------- - --------------------- + I*|--------------------------- - -----------------------||*|- + 3 / --- + -------- + -----------------------| |3 ________________ 2 | 2 ________________|| |3 ________________ 2 | 2 ________________|| |3 \/ 27 3 ________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 935 \/ 9057 | / 935 \/ 9057 || | / 935 \/ 9057 | / 935 \/ 9057 || | / 935 \/ 9057 | | 9*3 / --- + -------- | 9*3 / --- + -------- || | 9*3 / --- + -------- | 9*3 / --- + -------- || | 9*3 / --- + -------- | \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 \ \/ 27 3 // \ \/ 27 3 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} - \frac{26}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + i \left(- \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} - \frac{26}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}{2} + \frac{26 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}}\right)\right) \left(\frac{52}{9 \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}} + \frac{5}{3} + \sqrt[3]{\frac{\sqrt{9057}}{3} + \frac{935}{27}}\right)$$
=
45
$$45$$
45
Respuesta numérica
[src]
x1 = -1.07112748277456 + 2.27006984051383*i
x2 = 7.14225496554912
x3 = -1.07112748277456 - 2.27006984051383*i
x3 = -1.07112748277456 - 2.27006984051383*i