Sr Examen
6/x^2-1-2/x-1=2-x+4\x+1 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
6 2 4 -- - 1 - - - 1 = 2 - x + - + 1 2 x x x
$$\left(\left(-1 + \frac{6}{x^{2}}\right) - \frac{2}{x}\right) - 1 = \left(\left(2 - x\right) + \frac{4}{x}\right) + 1$$
Respuesta rápida
[src]
/ 1 \ / / 1 \ \
43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\ / / _____\\
| _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 || | |9*\/ 586 ||
|/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------|| |atan|---------||
|| 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /| _____ | \ 179 /|
||- - - -------|*3 / --- + --------- | | ||- - - -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------| \/ 129 *sin|---------------|
5 \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 / \ 3 /
x1 = - + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- - ----------------------------| - --------------------------- + ----------------------------
3 9 \ 9 6 6 / 6 6
$$x_{1} = - \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} + \frac{\sqrt{129} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{5}{3} + i \left(- \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9}\right)$$
/ 1 \ / / 1 \ \
43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\ / / _____\\
| _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 || | |9*\/ 586 ||
|/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------|| |atan|---------||
|| 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /| _____ | \ 179 /|
||- - + -------|*3 / --- + --------- | | ||- - + -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------| \/ 129 *sin|---------------|
5 \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 / \ 3 /
x2 = - + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- + ----------------------------| - --------------------------- - ----------------------------
3 9 \ 9 6 6 / 6 6
$$x_{2} = \frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{129} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{5}{3} + i \left(\frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6}\right)$$
_________________
/ _____
5 / 179 I*\/ 586 43
x3 = - + 3 / --- + --------- + ------------------------
3 \/ 27 3 _________________
/ _____
/ 179 I*\/ 586
9*3 / --- + ---------
\/ 27 3
$$x_{3} = \frac{5}{3} + \frac{43}{9 \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}} + \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}$$
x3 = 5/3 + 43/(9*(179/27 + sqrt(586)*i/3)^(1/3)) + (179/27 + sqrt(586)*i/3)^(1/3)
Suma y producto de raíces
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suma
/ 1 \ / / 1 \ \ / 1 \ / / 1 \ \
43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\ / / _____\\ 43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\ / / _____\\
| _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 || | |9*\/ 586 || | _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 || | |9*\/ 586 ||
|/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------|| |atan|---------|| |/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------|| |atan|---------||
|| 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /| _____ | \ 179 /| || 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /| _____ | \ 179 /| _________________
||- - - -------|*3 / --- + --------- | | ||- - - -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------| \/ 129 *sin|---------------| ||- - + -------|*3 / --- + --------- | | ||- - + -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------| \/ 129 *sin|---------------| / _____
5 \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 / \ 3 / 5 \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 / \ 3 / 5 / 179 I*\/ 586 43
- + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- - ----------------------------| - --------------------------- + ---------------------------- + - + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- + ----------------------------| - --------------------------- - ---------------------------- + - + 3 / --- + --------- + ------------------------
3 9 \ 9 6 6 / 6 6 3 9 \ 9 6 6 / 6 6 3 \/ 27 3 _________________
/ _____
/ 179 I*\/ 586
9*3 / --- + ---------
\/ 27 3
$$\left(\frac{5}{3} + \frac{43}{9 \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}} + \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}\right) + \left(\left(\frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{129} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{5}{3} + i \left(\frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6}\right)\right) + \left(- \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} + \frac{\sqrt{129} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{5}{3} + i \left(- \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9}\right)\right)\right)$$
=
/ 1 \ / 1 \ / / 1 \ \ / / 1 \ \
43*re|--------------------------------------| 43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\
| _________________| | _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 ||
|/ ___\ / _____ | |/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------||
_________________ || 1 I*\/ 3 | / 179 I*\/ 586 | || 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /|
/ _____ ||- - + -------|*3 / --- + --------- | ||- - - -------|*3 / --- + --------- | | ||- - + -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| | ||- - - -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------|
/ 179 I*\/ 586 43 \\ 2 2 / \/ 27 3 / \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 /
5 + 3 / --- + --------- + ------------------------ + --------------------------------------------- + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- + ----------------------------| + I*|--------------------------------------------- - --------------------------- - ----------------------------| - ---------------------------
\/ 27 3 _________________ 9 9 \ 9 6 6 / \ 9 6 6 / 3
/ _____
/ 179 I*\/ 586
9*3 / --- + ---------
\/ 27 3
$$- \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{3} + \frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} + \frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} + 5 + \frac{43}{9 \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}} + i \left(- \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9}\right) + i \left(\frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6}\right) + \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}$$
producto
/ / 1 \ / / 1 \ \ \ / / 1 \ / / 1 \ \ \
| 43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\ / / _____\\| | 43*re|--------------------------------------| |43*im|--------------------------------------| / / _____\\ / / _____\\| / / _____\\ / / _____\\|
| | _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 || | |9*\/ 586 ||| | | _________________| | | _________________| | |9*\/ 586 || | |9*\/ 586 ||| | |9*\/ 586 || | |9*\/ 586 |||
| |/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------|| |atan|---------||| | |/ ___\ / _____ | | |/ ___\ / _____ | |atan|---------|| |atan|---------||| |atan|---------|| |atan|---------|||
| || 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /| _____ | \ 179 /|| | || 1 I*\/ 3 | / 179 I*\/ 586 | | || 1 I*\/ 3 | / 179 I*\/ 586 | ____ | \ 179 /| _____ | \ 179 /|| ____ | \ 179 /| _____ | \ 179 /|| / _________________ \
| ||- - - -------|*3 / --- + --------- | | ||- - - -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------| \/ 129 *sin|---------------|| | ||- - + -------|*3 / --- + --------- | | ||- - + -------|*3 / --- + --------- | \/ 43 *sin|---------------| \/ 129 *cos|---------------|| \/ 43 *cos|---------------| \/ 129 *sin|---------------|| | / _____ |
|5 \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 / \ 3 /| |5 \\ 2 2 / \/ 27 3 / | \\ 2 2 / \/ 27 3 / \ 3 / \ 3 /| \ 3 / \ 3 /| |5 / 179 I*\/ 586 43 |
|- + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- - ----------------------------| - --------------------------- + ----------------------------|*|- + --------------------------------------------- + I*|--------------------------------------------- - --------------------------- + ----------------------------| - --------------------------- - ----------------------------|*|- + 3 / --- + --------- + ------------------------|
\3 9 \ 9 6 6 / 6 6 / \3 9 \ 9 6 6 / 6 6 / |3 \/ 27 3 _________________|
| / _____ |
| / 179 I*\/ 586 |
| 9*3 / --- + --------- |
\ \/ 27 3 /
$$\left(- \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} + \frac{\sqrt{129} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{5}{3} + i \left(- \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9}\right)\right) \left(\frac{43 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} - \frac{\sqrt{129} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{5}{3} + i \left(\frac{43 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}}\right)}}{9} - \frac{\sqrt{43} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6} + \frac{\sqrt{129} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}}{6}\right)\right) \left(\frac{5}{3} + \frac{43}{9 \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}} + \sqrt[3]{\frac{179}{27} + \frac{\sqrt{586} i}{3}}\right)$$
=
/ / / _____\\ / / _____\\\
| | |9*\/ 586 || | |9*\/ 586 |||
/ ___________________ / ___________________\\ | |2*atan|---------|| |atan|---------|||
| 3 / _____ | 3 / _____ || | | \ 179 /| ____ | \ 179 /||
\43 + \/ 179 + 9*I*\/ 586 *\5 + \/ 179 + 9*I*\/ 586 //*|-18 + 86*cos|-----------------| - 10*\/ 43 *cos|---------------||
\ \ 3 / \ 3 //
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___________________
3 / _____
27*\/ 179 + 9*I*\/ 586
$$\frac{\left(43 + \left(5 + \sqrt[3]{179 + 9 \sqrt{586} i}\right) \sqrt[3]{179 + 9 \sqrt{586} i}\right) \left(- 10 \sqrt{43} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)} - 18 + 86 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{9 \sqrt{586}}{179} \right)}}{3} \right)}\right)}{27 \sqrt[3]{179 + 9 \sqrt{586} i}}$$
(43 + (179 + 9*i*sqrt(586))^(1/3)*(5 + (179 + 9*i*sqrt(586))^(1/3)))*(-18 + 86*cos(2*atan(9*sqrt(586)/179)/3) - 10*sqrt(43)*cos(atan(9*sqrt(586)/179)/3))/(27*(179 + 9*i*sqrt(586))^(1/3))
Respuesta numérica
[src]
x1 = 0.673191369894797 - 0.e-22*i
x2 = -1.52347627148141 + 0.e-22*i
x3 = 5.85028490158661 - 0.e-20*i
x3 = 5.85028490158661 - 0.e-20*i