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Expresiones idénticas
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Expresiones semejantes
- x^3+25*x+19=0
- x^3-25*x-19=0
x^3-25*x+19=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -25$$
$$v = \frac{d}{a}$$
$$v = 19$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -25$$
$$x_{1} x_{2} x_{3} = 19$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -25$$
$$v = \frac{d}{a}$$
$$v = 19$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -25$$
$$x_{1} x_{2} x_{3} = 19$$
Suma y producto de raíces
[src]
suma
/ / ________\\ / / / ________\\ / / ________\\\ / / ________\\ / / ________\\ / / / ________\\ / / ________\\\ / / ________\\
| |\/ 158259 || | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 || | |\/ 158259 || | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 || ______________________
|atan|----------|| | |atan|----------|| |atan|----------||| |atan|----------|| |atan|----------|| | |atan|----------|| |atan|----------||| |atan|----------|| / ________
| \ 171 /| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /| | \ 171 /| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /| / 513 3*I*\/ 158259
5*sin|----------------| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------| 5*sin|----------------| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------| 3 / --- + --------------
/ 1 \ \ 3 / | / 1 \ \ 3 / \ 3 /| \ 3 / / 1 \ \ 3 / | / 1 \ \ 3 / \ 3 /| \ 3 / 25 \/ 2 2
- 25*re|-------------------------------------------| - ----------------------- + I*|- 25*im|-------------------------------------------| + ----------------------- + -----------------------------| + ----------------------------- + - 25*re|-------------------------------------------| + ----------------------- + I*|- 25*im|-------------------------------------------| - ----------------------- + -----------------------------| + ----------------------------- + - --------------------------- - ---------------------------
| ______________________| 2 | | ______________________| 2 6 | 6 | ______________________| 2 | | ______________________| 2 6 | 6 ______________________ 3
|/ ___\ / ________ | | |/ ___\ / ________ | | |/ ___\ / ________ | | |/ ___\ / ________ | | / ________
|| 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | / 513 3*I*\/ 158259
||- - - -------|*3 / --- + -------------- | | ||- - - -------|*3 / --- + -------------- | | ||- - + -------|*3 / --- + -------------- | | ||- - + -------|*3 / --- + -------------- | | 3 / --- + --------------
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / \/ 2 2
$$\left(\left(\frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + i \left(- \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)}\right)\right) + \left(- \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + i \left(- 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}{3} - \frac{25}{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)$$
=
/ / / ________\\ / / ________\\\ / / / ________\\ / / ________\\\ / / ________\\
______________________ | | |\/ 158259 || | |\/ 158259 ||| | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 ||
/ ________ | |atan|----------|| |atan|----------||| | |atan|----------|| |atan|----------||| |atan|----------||
/ 513 3*I*\/ 158259 | | \ 171 /| ___ | \ 171 /|| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /|
3 / --- + -------------- | 5*cos|----------------| 5*\/ 3 *sin|----------------|| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------|
25 / 1 \ / 1 \ \/ 2 2 | / 1 \ \ 3 / \ 3 /| | / 1 \ \ 3 / \ 3 /| \ 3 /
- --------------------------- - 25*re|-------------------------------------------| - 25*re|-------------------------------------------| - --------------------------- + I*|- 25*im|-------------------------------------------| - ----------------------- + -----------------------------| + I*|- 25*im|-------------------------------------------| + ----------------------- + -----------------------------| + -----------------------------
______________________ | ______________________| | ______________________| 3 | | ______________________| 2 6 | | | ______________________| 2 6 | 3
/ ________ |/ ___\ / ________ | |/ ___\ / ________ | | |/ ___\ / ________ | | | |/ ___\ / ________ | |
/ 513 3*I*\/ 158259 || 1 I*\/ 3 | / 513 3*I*\/ 158259 | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | |
3 / --- + -------------- ||- - + -------|*3 / --- + -------------- | ||- - - -------|*3 / --- + -------------- | | ||- - + -------|*3 / --- + -------------- | | | ||- - - -------|*3 / --- + -------------- | |
\/ 2 2 \\ 2 2 / \/ 2 2 / \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / \ \\ 2 2 / \/ 2 2 / /
$$- 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{3} - \frac{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}{3} + i \left(- \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)}\right) + i \left(- 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2}\right) - \frac{25}{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}$$
producto
/ / / ________\\ / / / ________\\ / / ________\\\ / / ________\\\ / / / ________\\ / / / ________\\ / / ________\\\ / / ________\\\ | | |\/ 158259 || | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 ||| | | |\/ 158259 || | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 ||| / ______________________\ | |atan|----------|| | |atan|----------|| |atan|----------||| |atan|----------||| | |atan|----------|| | |atan|----------|| |atan|----------||| |atan|----------||| | / ________ | | | \ 171 /| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /|| | | \ 171 /| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /|| | / 513 3*I*\/ 158259 | | 5*sin|----------------| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------|| | 5*sin|----------------| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------|| | 3 / --- + -------------- | | / 1 \ \ 3 / | / 1 \ \ 3 / \ 3 /| \ 3 /| | / 1 \ \ 3 / | / 1 \ \ 3 / \ 3 /| \ 3 /| | 25 \/ 2 2 | |- 25*re|-------------------------------------------| - ----------------------- + I*|- 25*im|-------------------------------------------| + ----------------------- + -----------------------------| + -----------------------------|*|- 25*re|-------------------------------------------| + ----------------------- + I*|- 25*im|-------------------------------------------| - ----------------------- + -----------------------------| + -----------------------------|*|- --------------------------- - ---------------------------| | | ______________________| 2 | | ______________________| 2 6 | 6 | | | ______________________| 2 | | ______________________| 2 6 | 6 | | ______________________ 3 | | |/ ___\ / ________ | | |/ ___\ / ________ | | | | |/ ___\ / ________ | | |/ ___\ / ________ | | | | / ________ | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | | | | / 513 3*I*\/ 158259 | | ||- - - -------|*3 / --- + -------------- | | ||- - - -------|*3 / --- + -------------- | | | | ||- - + -------|*3 / --- + -------------- | | ||- - + -------|*3 / --- + -------------- | | | | 3 / --- + -------------- | \ \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / / \ \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / / \ \/ 2 2 /
$$\left(- \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + i \left(- 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2}\right)\right) \left(\frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + i \left(- \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)}\right)\right) \left(- \frac{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}{3} - \frac{25}{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)$$
=
/ / / ________\\\
| | |\/ 158259 |||
| |2*atan|----------|||
| | \ 171 /||
| cos|------------------|| / 2/3 \
2/3 |1 \ 3 /| |/ ________\ 3 ___|
50*3 *|- - -----------------------|*\\1026 + 6*I*\/ 158259 / + 150*\/ 2 /
\4 2 /
------------------------------------------------------------------------------
____________________
3 / ________
27*\/ 171 + I*\/ 158259
$$\frac{50 \cdot 3^{\frac{2}{3}} \left(\frac{1}{4} - \frac{\cos{\left(\frac{2 \operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2}\right) \left(150 \sqrt[3]{2} + \left(1026 + 6 \sqrt{158259} i\right)^{\frac{2}{3}}\right)}{27 \sqrt[3]{171 + \sqrt{158259} i}}$$
50*3^(2/3)*(1/4 - cos(2*atan(sqrt(158259)/171)/3)/2)*((1026 + 6*i*sqrt(158259))^(2/3) + 150*2^(1/3))/(27*(171 + i*sqrt(158259))^(1/3))
Respuesta rápida
[src]
/ / ________\\ / / / ________\\ / / ________\\\ / / ________\\
| |\/ 158259 || | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 ||
|atan|----------|| | |atan|----------|| |atan|----------||| |atan|----------||
| \ 171 /| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /|
5*sin|----------------| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------|
/ 1 \ \ 3 / | / 1 \ \ 3 / \ 3 /| \ 3 /
x1 = - 25*re|-------------------------------------------| - ----------------------- + I*|- 25*im|-------------------------------------------| + ----------------------- + -----------------------------| + -----------------------------
| ______________________| 2 | | ______________________| 2 6 | 6
|/ ___\ / ________ | | |/ ___\ / ________ | |
|| 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | |
||- - - -------|*3 / --- + -------------- | | ||- - - -------|*3 / --- + -------------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{1} = - \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + i \left(- 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2}\right)$$
/ / ________\\ / / / ________\\ / / ________\\\ / / ________\\
| |\/ 158259 || | | |\/ 158259 || | |\/ 158259 ||| | |\/ 158259 ||
|atan|----------|| | |atan|----------|| |atan|----------||| |atan|----------||
| \ 171 /| | | \ 171 /| ___ | \ 171 /|| ___ | \ 171 /|
5*sin|----------------| | 5*cos|----------------| 5*\/ 3 *sin|----------------|| 5*\/ 3 *cos|----------------|
/ 1 \ \ 3 / | / 1 \ \ 3 / \ 3 /| \ 3 /
x2 = - 25*re|-------------------------------------------| + ----------------------- + I*|- 25*im|-------------------------------------------| - ----------------------- + -----------------------------| + -----------------------------
| ______________________| 2 | | ______________________| 2 6 | 6
|/ ___\ / ________ | | |/ ___\ / ________ | |
|| 1 I*\/ 3 | / 513 3*I*\/ 158259 | | || 1 I*\/ 3 | / 513 3*I*\/ 158259 | |
||- - + -------|*3 / --- + -------------- | | ||- - + -------|*3 / --- + -------------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{2} = \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)} + i \left(- \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{2} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{158259}}{171} \right)}}{3} \right)}}{6} - 25 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}\right)}\right)$$
______________________
/ ________
/ 513 3*I*\/ 158259
3 / --- + --------------
25 \/ 2 2
x3 = - --------------------------- - ---------------------------
______________________ 3
/ ________
/ 513 3*I*\/ 158259
3 / --- + --------------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}{3} - \frac{25}{\sqrt[3]{\frac{513}{2} + \frac{3 \sqrt{158259} i}{2}}}$$
x3 = -(513/2 + 3*sqrt(158259)*i/2)^(1/3)/3 - 25/(513/2 + 3*sqrt(158259)*i/2)^(1/3)
Respuesta numérica
[src]
x1 = 0.778902032363749
x2 = -5.34374044232411
x3 = 4.56483840996036
x3 = 4.56483840996036