(19/50)^3*x^3+19*2*(19/50)^3*x^2/50-37*34577/(10*100*10^3*5/8)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
( ( 19 50 ) 3 x 3 + 38 ( 19 50 ) 3 x 2 50 ) − 1279349 5 8 ⋅ 1000000 = 0 \left(\left(\frac{19}{50}\right)^{3} x^{3} + \frac{38 \left(\frac{19}{50}\right)^{3} x^{2}}{50}\right) - \frac{1279349}{\frac{5}{8} \cdot 1000000} = 0 ( ( 50 19 ) 3 x 3 + 50 38 ( 50 19 ) 3 x 2 ) − 8 5 ⋅ 1000000 1279349 = 0 de
a x 3 + b x 2 + c x + d = 0 a x^{3} + b x^{2} + c x + d = 0 a x 3 + b x 2 + c x + d = 0 como ecuación cúbica reducida
x 3 + b x 2 a + c x a + d a = 0 x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0 x 3 + a b x 2 + a c x + a d = 0 x 3 + 19 x 2 25 − 1279349 34295 = 0 x^{3} + \frac{19 x^{2}}{25} - \frac{1279349}{34295} = 0 x 3 + 25 19 x 2 − 34295 1279349 = 0 p x 2 + q x + v + x 3 = 0 p x^{2} + q x + v + x^{3} = 0 p x 2 + q x + v + x 3 = 0 donde
p = b a p = \frac{b}{a} p = a b p = 19 25 p = \frac{19}{25} p = 25 19 q = c a q = \frac{c}{a} q = a c q = 0 q = 0 q = 0 v = d a v = \frac{d}{a} v = a d v = − 1279349 34295 v = - \frac{1279349}{34295} v = − 34295 1279349 Fórmulas de Cardano-Vieta
x 1 + x 2 + x 3 = − p x_{1} + x_{2} + x_{3} = - p 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_{1} x_{3} + x_{2} x_{3} = q 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} = v x 1 x 2 x 3 = v x 1 + x 2 + x 3 = − 19 25 x_{1} + x_{2} + x_{3} = - \frac{19}{25} x 1 + x 2 + x 3 = − 25 19 x 1 x 2 + x 1 x 3 + x 2 x 3 = 0 x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0 x 1 x 2 + x 1 x 3 + x 2 x 3 = 0 x 1 x 2 x 3 = − 1279349 34295 x_{1} x_{2} x_{3} = - \frac{1279349}{34295} x 1 x 2 x 3 = − 34295 1279349
Gráfica
-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 -200 200
Suma y producto de raíces
[src]
________________________________________ / ________________________________________ \ ________________________________________ / ________________________________________ \
/ _____________________ | / _____________________ | / _____________________ | / _____________________ |
/ 107850980113 \/ 2067880010324452485 | ___ / 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 | ___ / 107850980113 \/ 2067880010324452485 | ________________________________________
3 / ------------ + ----------------------- | \/ 3 *3 / ------------ + ----------------------- ___ | 3 / ------------ + ----------------------- |\/ 3 *3 / ------------ + ----------------------- ___ | / _____________________
19 361 \/ 5787281250 77163750 | \/ 5787281250 77163750 361*\/ 3 | 19 361 \/ 5787281250 77163750 | \/ 5787281250 77163750 361*\/ 3 | 19 / 107850980113 \/ 2067880010324452485 361
- -- - --------------------------------------------------- - --------------------------------------------- + I*|- --------------------------------------------------- + ---------------------------------------------------| + - -- - --------------------------------------------------- - --------------------------------------------- + I*|--------------------------------------------------- - ---------------------------------------------------| + - -- + 3 / ------------ + ----------------------- + --------------------------------------------------
75 ________________________________________ 2 | 2 ________________________________________| 75 ________________________________________ 2 | 2 ________________________________________| 75 \/ 5787281250 77163750 ________________________________________
/ _____________________ | / _____________________ | / _____________________ | / _____________________ | / _____________________
/ 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485
11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- | 5625*3 / ------------ + -----------------------
\/ 5787281250 77163750 \ \/ 5787281250 77163750 / \/ 5787281250 77163750 \ \/ 5787281250 77163750 / \/ 5787281250 77163750
( − 19 75 + 361 5625 2067880010324452485 77163750 + 107850980113 5787281250 3 + 2067880010324452485 77163750 + 107850980113 5787281250 3 ) + ( ( − 2067880010324452485 77163750 + 107850980113 5787281250 3 2 − 19 75 − 361 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + i ( − 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 + 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 ) ) + ( − 2067880010324452485 77163750 + 107850980113 5787281250 3 2 − 19 75 − 361 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + i ( − 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 ) ) ) \left(- \frac{19}{75} + \frac{361}{5625 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} - \frac{19}{75} - \frac{361}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} + \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} - \frac{19}{75} - \frac{361}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + i \left(- \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2}\right)\right)\right) − 75 19 + 5625 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + 3 77163750 2067880010324452485 + 5787281250 107850980113 + − 2 3 77163750 2067880010324452485 + 5787281250 107850980113 − 75 19 − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + i − 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113 + 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 + − 2 3 77163750 2067880010324452485 + 5787281250 107850980113 − 75 19 − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + i − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 + 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113
/ ________________________________________ \ / ________________________________________ \
| / _____________________ | | / _____________________ |
| ___ / 107850980113 \/ 2067880010324452485 | | ___ / 107850980113 \/ 2067880010324452485 |
|\/ 3 *3 / ------------ + ----------------------- ___ | | \/ 3 *3 / ------------ + ----------------------- ___ |
19 | \/ 5787281250 77163750 361*\/ 3 | | \/ 5787281250 77163750 361*\/ 3 |
- -- + I*|--------------------------------------------------- - ---------------------------------------------------| + I*|- --------------------------------------------------- + ---------------------------------------------------|
25 | 2 ________________________________________| | 2 ________________________________________|
| / _____________________ | | / _____________________ |
| / 107850980113 \/ 2067880010324452485 | | / 107850980113 \/ 2067880010324452485 |
| 11250*3 / ------------ + ----------------------- | | 11250*3 / ------------ + ----------------------- |
\ \/ 5787281250 77163750 / \ \/ 5787281250 77163750 /
− 19 25 + i ( − 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 + 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 ) + i ( − 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 ) - \frac{19}{25} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} + \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}\right) + i \left(- \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2}\right) − 25 19 + i − 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113 + 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 + i − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 + 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113
/ ________________________________________ / ________________________________________ \\ / ________________________________________ / ________________________________________ \\
| / _____________________ | / _____________________ || | / _____________________ | / _____________________ ||
| / 107850980113 \/ 2067880010324452485 | ___ / 107850980113 \/ 2067880010324452485 || | / 107850980113 \/ 2067880010324452485 | ___ / 107850980113 \/ 2067880010324452485 || / ________________________________________ \
| 3 / ------------ + ----------------------- | \/ 3 *3 / ------------ + ----------------------- ___ || | 3 / ------------ + ----------------------- |\/ 3 *3 / ------------ + ----------------------- ___ || | / _____________________ |
| 19 361 \/ 5787281250 77163750 | \/ 5787281250 77163750 361*\/ 3 || | 19 361 \/ 5787281250 77163750 | \/ 5787281250 77163750 361*\/ 3 || | 19 / 107850980113 \/ 2067880010324452485 361 |
|- -- - --------------------------------------------------- - --------------------------------------------- + I*|- --------------------------------------------------- + ---------------------------------------------------||*|- -- - --------------------------------------------------- - --------------------------------------------- + I*|--------------------------------------------------- - ---------------------------------------------------||*|- -- + 3 / ------------ + ----------------------- + --------------------------------------------------|
| 75 ________________________________________ 2 | 2 ________________________________________|| | 75 ________________________________________ 2 | 2 ________________________________________|| | 75 \/ 5787281250 77163750 ________________________________________|
| / _____________________ | / _____________________ || | / _____________________ | / _____________________ || | / _____________________ |
| / 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 || | / 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 || | / 107850980113 \/ 2067880010324452485 |
| 11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- || | 11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- || | 5625*3 / ------------ + ----------------------- |
\ \/ 5787281250 77163750 \ \/ 5787281250 77163750 // \ \/ 5787281250 77163750 \ \/ 5787281250 77163750 // \ \/ 5787281250 77163750 /
( − 2067880010324452485 77163750 + 107850980113 5787281250 3 2 − 19 75 − 361 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + i ( − 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 ) ) ( − 2067880010324452485 77163750 + 107850980113 5787281250 3 2 − 19 75 − 361 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + i ( − 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 + 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 ) ) ( − 19 75 + 361 5625 2067880010324452485 77163750 + 107850980113 5787281250 3 + 2067880010324452485 77163750 + 107850980113 5787281250 3 ) \left(- \frac{\sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} - \frac{19}{75} - \frac{361}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + i \left(- \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} - \frac{19}{75} - \frac{361}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} + \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}\right)\right) \left(- \frac{19}{75} + \frac{361}{5625 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}\right) − 2 3 77163750 2067880010324452485 + 5787281250 107850980113 − 75 19 − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + i − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 + 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113 − 2 3 77163750 2067880010324452485 + 5787281250 107850980113 − 75 19 − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + i − 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113 + 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 − 75 19 + 5625 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + 3 77163750 2067880010324452485 + 5787281250 107850980113
1279349 34295 \frac{1279349}{34295} 34295 1279349
________________________________________ / ________________________________________ \
/ _____________________ | / _____________________ |
/ 107850980113 \/ 2067880010324452485 | ___ / 107850980113 \/ 2067880010324452485 |
3 / ------------ + ----------------------- | \/ 3 *3 / ------------ + ----------------------- ___ |
19 361 \/ 5787281250 77163750 | \/ 5787281250 77163750 361*\/ 3 |
x1 = - -- - --------------------------------------------------- - --------------------------------------------- + I*|- --------------------------------------------------- + ---------------------------------------------------|
75 ________________________________________ 2 | 2 ________________________________________|
/ _____________________ | / _____________________ |
/ 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 |
11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- |
\/ 5787281250 77163750 \ \/ 5787281250 77163750 /
x 1 = − 2067880010324452485 77163750 + 107850980113 5787281250 3 2 − 19 75 − 361 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + i ( − 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 + 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 ) x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} - \frac{19}{75} - \frac{361}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} + \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}\right) x 1 = − 2 3 77163750 2067880010324452485 + 5787281250 107850980113 − 75 19 − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + i − 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113 + 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3
________________________________________ / ________________________________________ \
/ _____________________ | / _____________________ |
/ 107850980113 \/ 2067880010324452485 | ___ / 107850980113 \/ 2067880010324452485 |
3 / ------------ + ----------------------- |\/ 3 *3 / ------------ + ----------------------- ___ |
19 361 \/ 5787281250 77163750 | \/ 5787281250 77163750 361*\/ 3 |
x2 = - -- - --------------------------------------------------- - --------------------------------------------- + I*|--------------------------------------------------- - ---------------------------------------------------|
75 ________________________________________ 2 | 2 ________________________________________|
/ _____________________ | / _____________________ |
/ 107850980113 \/ 2067880010324452485 | / 107850980113 \/ 2067880010324452485 |
11250*3 / ------------ + ----------------------- | 11250*3 / ------------ + ----------------------- |
\/ 5787281250 77163750 \ \/ 5787281250 77163750 /
x 2 = − 2067880010324452485 77163750 + 107850980113 5787281250 3 2 − 19 75 − 361 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + i ( − 361 3 11250 2067880010324452485 77163750 + 107850980113 5787281250 3 + 3 2067880010324452485 77163750 + 107850980113 5787281250 3 2 ) x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2} - \frac{19}{75} - \frac{361}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + i \left(- \frac{361 \sqrt{3}}{11250 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}}{2}\right) x 2 = − 2 3 77163750 2067880010324452485 + 5787281250 107850980113 − 75 19 − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + i − 11250 3 77163750 2067880010324452485 + 5787281250 107850980113 361 3 + 2 3 3 77163750 2067880010324452485 + 5787281250 107850980113
________________________________________
/ _____________________
19 / 107850980113 \/ 2067880010324452485 361
x3 = - -- + 3 / ------------ + ----------------------- + --------------------------------------------------
75 \/ 5787281250 77163750 ________________________________________
/ _____________________
/ 107850980113 \/ 2067880010324452485
5625*3 / ------------ + -----------------------
\/ 5787281250 77163750
x 3 = − 19 75 + 361 5625 2067880010324452485 77163750 + 107850980113 5787281250 3 + 2067880010324452485 77163750 + 107850980113 5787281250 3 x_{3} = - \frac{19}{75} + \frac{361}{5625 \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}}} + \sqrt[3]{\frac{\sqrt{2067880010324452485}}{77163750} + \frac{107850980113}{5787281250}} x 3 = − 75 19 + 5625 3 77163750 2067880010324452485 + 5787281250 107850980113 361 + 3 77163750 2067880010324452485 + 5787281250 107850980113
x3 = -19/75 + 361/(5625*(sqrt(2067880010324452485)/77163750 + 107850980113/5787281250)^(1/3)) + (sqrt(2067880010324452485)/77163750 + 107850980113/5787281250)^(1/3)
x1 = -1.9331192361471 + 2.87619691685779*i
x3 = -1.9331192361471 - 2.87619691685779*i
x3 = -1.9331192361471 - 2.87619691685779*i