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Expresiones semejantes
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x^3-7x^2-9*x-13=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - 7*x - 9*x - 13 = 0
$$\left(- 9 x + \left(x^{3} - 7 x^{2}\right)\right) - 13 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -13$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 7$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -13$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -13$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 7$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -13$$
Respuesta rápida
[src]
__________________ / __________________ \
/ ______ | / ______ |
/ 802 2*\/ 5673 | ___ / 802 2*\/ 5673 |
3 / --- + ---------- | \/ 3 *3 / --- + ---------- ___ |
7 38 \/ 27 9 | \/ 27 9 38*\/ 3 |
x1 = - - ------------------------- - ----------------------- + I*|- ----------------------------- + -------------------------|
3 __________________ 2 | 2 __________________|
/ ______ | / ______ |
/ 802 2*\/ 5673 | / 802 2*\/ 5673 |
9*3 / --- + ---------- | 9*3 / --- + ---------- |
\/ 27 9 \ \/ 27 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} - \frac{38}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{7}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} + \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}\right)$$
__________________ / __________________ \
/ ______ | / ______ |
/ 802 2*\/ 5673 | ___ / 802 2*\/ 5673 |
3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ |
7 38 \/ 27 9 | \/ 27 9 38*\/ 3 |
x2 = - - ------------------------- - ----------------------- + I*|----------------------------- - -------------------------|
3 __________________ 2 | 2 __________________|
/ ______ | / ______ |
/ 802 2*\/ 5673 | / 802 2*\/ 5673 |
9*3 / --- + ---------- | 9*3 / --- + ---------- |
\/ 27 9 \ \/ 27 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} - \frac{38}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{7}{3} + i \left(- \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2}\right)$$
__________________
/ ______
7 / 802 2*\/ 5673 76
x3 = - + 3 / --- + ---------- + -------------------------
3 \/ 27 9 __________________
/ ______
/ 802 2*\/ 5673
9*3 / --- + ----------
\/ 27 9
$$x_{3} = \frac{7}{3} + \frac{76}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}$$
x3 = 7/3 + 76/(9*(2*sqrt(5673)/9 + 802/27)^(1/3)) + (2*sqrt(5673)/9 + 802/27)^(1/3)
Suma y producto de raíces
[src]
suma
__________________ / __________________ \ __________________ / __________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 802 2*\/ 5673 | ___ / 802 2*\/ 5673 | / 802 2*\/ 5673 | ___ / 802 2*\/ 5673 | __________________
3 / --- + ---------- | \/ 3 *3 / --- + ---------- ___ | 3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ | / ______
7 38 \/ 27 9 | \/ 27 9 38*\/ 3 | 7 38 \/ 27 9 | \/ 27 9 38*\/ 3 | 7 / 802 2*\/ 5673 76
- - ------------------------- - ----------------------- + I*|- ----------------------------- + -------------------------| + - - ------------------------- - ----------------------- + I*|----------------------------- - -------------------------| + - + 3 / --- + ---------- + -------------------------
3 __________________ 2 | 2 __________________| 3 __________________ 2 | 2 __________________| 3 \/ 27 9 __________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 802 2*\/ 5673 | / 802 2*\/ 5673 | / 802 2*\/ 5673 | / 802 2*\/ 5673 | / 802 2*\/ 5673
9*3 / --- + ---------- | 9*3 / --- + ---------- | 9*3 / --- + ---------- | 9*3 / --- + ---------- | 9*3 / --- + ----------
\/ 27 9 \ \/ 27 9 / \/ 27 9 \ \/ 27 9 / \/ 27 9
$$\left(\frac{7}{3} + \frac{76}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} - \frac{38}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{7}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} + \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} - \frac{38}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{7}{3} + i \left(- \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2}\right)\right)\right)$$
=
/ __________________ \ / __________________ \
| / ______ | | / ______ |
| ___ / 802 2*\/ 5673 | | ___ / 802 2*\/ 5673 |
|\/ 3 *3 / --- + ---------- ___ | | \/ 3 *3 / --- + ---------- ___ |
| \/ 27 9 38*\/ 3 | | \/ 27 9 38*\/ 3 |
7 + I*|----------------------------- - -------------------------| + I*|- ----------------------------- + -------------------------|
| 2 __________________| | 2 __________________|
| / ______ | | / ______ |
| / 802 2*\/ 5673 | | / 802 2*\/ 5673 |
| 9*3 / --- + ---------- | | 9*3 / --- + ---------- |
\ \/ 27 9 / \ \/ 27 9 /
$$7 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} + \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}\right) + i \left(- \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2}\right)$$
producto
/ __________________ / __________________ \\ / __________________ / __________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 802 2*\/ 5673 | ___ / 802 2*\/ 5673 || | / 802 2*\/ 5673 | ___ / 802 2*\/ 5673 || / __________________ \ | 3 / --- + ---------- | \/ 3 *3 / --- + ---------- ___ || | 3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ || | / ______ | |7 38 \/ 27 9 | \/ 27 9 38*\/ 3 || |7 38 \/ 27 9 | \/ 27 9 38*\/ 3 || |7 / 802 2*\/ 5673 76 | |- - ------------------------- - ----------------------- + I*|- ----------------------------- + -------------------------||*|- - ------------------------- - ----------------------- + I*|----------------------------- - -------------------------||*|- + 3 / --- + ---------- + -------------------------| |3 __________________ 2 | 2 __________________|| |3 __________________ 2 | 2 __________________|| |3 \/ 27 9 __________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 802 2*\/ 5673 | / 802 2*\/ 5673 || | / 802 2*\/ 5673 | / 802 2*\/ 5673 || | / 802 2*\/ 5673 | | 9*3 / --- + ---------- | 9*3 / --- + ---------- || | 9*3 / --- + ---------- | 9*3 / --- + ---------- || | 9*3 / --- + ---------- | \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 /
$$\left(- \frac{\sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} - \frac{38}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{7}{3} + i \left(- \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} - \frac{38}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \frac{7}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}{2} + \frac{38 \sqrt{3}}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}}\right)\right) \left(\frac{7}{3} + \frac{76}{9 \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}} + \sqrt[3]{\frac{2 \sqrt{5673}}{9} + \frac{802}{27}}\right)$$
=
13
$$13$$
13
Respuesta numérica
[src]
x1 = 8.27709117277217
x2 = -0.638545586386086 + 1.07835968354008*i
x3 = -0.638545586386086 - 1.07835968354008*i
x3 = -0.638545586386086 - 1.07835968354008*i