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Expresiones semejantes
- 16*x^3+12*x^2+24*x+15=0
- 16*x^3+12*x^2-24*x-15=0
- 16*x^3-12*x^2-24*x+15=0
16*x^3+12*x^2-24*x+15=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 16*x + 12*x - 24*x + 15 = 0
$$\left(- 24 x + \left(16 x^{3} + 12 x^{2}\right)\right) + 15 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 24 x + \left(16 x^{3} + 12 x^{2}\right)\right) + 15 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{3 x^{2}}{4} - \frac{3 x}{2} + \frac{15}{16} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{3}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{3}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{15}{16}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{3}{4}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{3}{2}$$
$$x_{1} x_{2} x_{3} = \frac{15}{16}$$
$$\left(- 24 x + \left(16 x^{3} + 12 x^{2}\right)\right) + 15 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{3 x^{2}}{4} - \frac{3 x}{2} + \frac{15}{16} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \frac{3}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{3}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{15}{16}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{3}{4}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{3}{2}$$
$$x_{1} x_{2} x_{3} = \frac{15}{16}$$
Respuesta rápida
[src]
__________________ / __________________\
/ ____ | / ____ |
/ 1161 27*\/ 70 | ___ / 1161 27*\/ 70 |
3 / ---- + --------- | ___ \/ 3 *3 / ---- + --------- |
1 \/ 64 16 27 | 27*\/ 3 \/ 64 16 |
x1 = - - + ----------------------- + -------------------------- + I*|- -------------------------- + -----------------------------|
4 6 __________________ | __________________ 6 |
/ ____ | / ____ |
/ 1161 27*\/ 70 | / 1161 27*\/ 70 |
32*3 / ---- + --------- | 32*3 / ---- + --------- |
\/ 64 16 \ \/ 64 16 /
$$x_{1} = - \frac{1}{4} + \frac{27}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + i \left(- \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6}\right)$$
__________________ / __________________ \
/ ____ | / ____ |
/ 1161 27*\/ 70 | ___ / 1161 27*\/ 70 |
3 / ---- + --------- | \/ 3 *3 / ---- + --------- ___ |
1 \/ 64 16 27 | \/ 64 16 27*\/ 3 |
x2 = - - + ----------------------- + -------------------------- + I*|- ----------------------------- + --------------------------|
4 6 __________________ | 6 __________________|
/ ____ | / ____ |
/ 1161 27*\/ 70 | / 1161 27*\/ 70 |
32*3 / ---- + --------- | 32*3 / ---- + --------- |
\/ 64 16 \ \/ 64 16 /
$$x_{2} = - \frac{1}{4} + \frac{27}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}\right)$$
__________________
/ ____
/ 1161 27*\/ 70
3 / ---- + ---------
1 27 \/ 64 16
x3 = - - - -------------------------- - -----------------------
4 __________________ 3
/ ____
/ 1161 27*\/ 70
16*3 / ---- + ---------
\/ 64 16
$$x_{3} = - \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{3} - \frac{27}{16 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} - \frac{1}{4}$$
x3 = -(27*sqrt(70)/16 + 1161/64)^(1/3)/3 - 27/(16*(27*sqrt(70)/16 + 1161/64)^(1/3)) - 1/4
Suma y producto de raíces
[src]
suma
__________________ / __________________\ __________________ / __________________ \ __________________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 1161 27*\/ 70 | ___ / 1161 27*\/ 70 | / 1161 27*\/ 70 | ___ / 1161 27*\/ 70 | / 1161 27*\/ 70
3 / ---- + --------- | ___ \/ 3 *3 / ---- + --------- | 3 / ---- + --------- | \/ 3 *3 / ---- + --------- ___ | 3 / ---- + ---------
1 \/ 64 16 27 | 27*\/ 3 \/ 64 16 | 1 \/ 64 16 27 | \/ 64 16 27*\/ 3 | 1 27 \/ 64 16
- - + ----------------------- + -------------------------- + I*|- -------------------------- + -----------------------------| + - - + ----------------------- + -------------------------- + I*|- ----------------------------- + --------------------------| + - - - -------------------------- - -----------------------
4 6 __________________ | __________________ 6 | 4 6 __________________ | 6 __________________| 4 __________________ 3
/ ____ | / ____ | / ____ | / ____ | / ____
/ 1161 27*\/ 70 | / 1161 27*\/ 70 | / 1161 27*\/ 70 | / 1161 27*\/ 70 | / 1161 27*\/ 70
32*3 / ---- + --------- | 32*3 / ---- + --------- | 32*3 / ---- + --------- | 32*3 / ---- + --------- | 16*3 / ---- + ---------
\/ 64 16 \ \/ 64 16 / \/ 64 16 \ \/ 64 16 / \/ 64 16
$$\left(- \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{3} - \frac{27}{16 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} - \frac{1}{4}\right) + \left(\left(- \frac{1}{4} + \frac{27}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}\right)\right) + \left(- \frac{1}{4} + \frac{27}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + i \left(- \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6}\right)\right)\right)$$
=
/ __________________\ / __________________ \
| / ____ | | / ____ |
| ___ / 1161 27*\/ 70 | | ___ / 1161 27*\/ 70 |
| ___ \/ 3 *3 / ---- + --------- | | \/ 3 *3 / ---- + --------- ___ |
3 | 27*\/ 3 \/ 64 16 | | \/ 64 16 27*\/ 3 |
- - + I*|- -------------------------- + -----------------------------| + I*|- ----------------------------- + --------------------------|
4 | __________________ 6 | | 6 __________________|
| / ____ | | / ____ |
| / 1161 27*\/ 70 | | / 1161 27*\/ 70 |
| 32*3 / ---- + --------- | | 32*3 / ---- + --------- |
\ \/ 64 16 / \ \/ 64 16 /
$$- \frac{3}{4} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}\right) + i \left(- \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6}\right)$$
producto
/ __________________ / __________________\\ / __________________ / __________________ \\ / __________________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 1161 27*\/ 70 | ___ / 1161 27*\/ 70 || | / 1161 27*\/ 70 | ___ / 1161 27*\/ 70 || | / 1161 27*\/ 70 | | 3 / ---- + --------- | ___ \/ 3 *3 / ---- + --------- || | 3 / ---- + --------- | \/ 3 *3 / ---- + --------- ___ || | 3 / ---- + --------- | | 1 \/ 64 16 27 | 27*\/ 3 \/ 64 16 || | 1 \/ 64 16 27 | \/ 64 16 27*\/ 3 || | 1 27 \/ 64 16 | |- - + ----------------------- + -------------------------- + I*|- -------------------------- + -----------------------------||*|- - + ----------------------- + -------------------------- + I*|- ----------------------------- + --------------------------||*|- - - -------------------------- - -----------------------| | 4 6 __________________ | __________________ 6 || | 4 6 __________________ | 6 __________________|| | 4 __________________ 3 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 1161 27*\/ 70 | / 1161 27*\/ 70 || | / 1161 27*\/ 70 | / 1161 27*\/ 70 || | / 1161 27*\/ 70 | | 32*3 / ---- + --------- | 32*3 / ---- + --------- || | 32*3 / ---- + --------- | 32*3 / ---- + --------- || | 16*3 / ---- + --------- | \ \/ 64 16 \ \/ 64 16 // \ \/ 64 16 \ \/ 64 16 // \ \/ 64 16 /
$$\left(- \frac{1}{4} + \frac{27}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + i \left(- \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6}\right)\right) \left(- \frac{1}{4} + \frac{27}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} + \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{6} + \frac{27 \sqrt{3}}{32 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}}{3} - \frac{27}{16 \sqrt[3]{\frac{27 \sqrt{70}}{16} + \frac{1161}{64}}} - \frac{1}{4}\right)$$
=
-15 ---- 16
$$- \frac{15}{16}$$
-15/16
Respuesta numérica
[src]
x1 = 0.545609560164386 - 0.459873587711779*i
x2 = 0.545609560164386 + 0.459873587711779*i
x3 = -1.84121912032877
x3 = -1.84121912032877