Sr Examen
x^3+2016*x^2=1008 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
$$p = \frac{b}{a}$$
$$p = 2016$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -1008$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2016$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -1008$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2016$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -1008$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2016$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -1008$$
Suma y producto de raíces
[src]
suma
/ / _________\\ / / / _________\\ / / _________\\\ / / _________\\ / / _________\\ / / / _________\\ / / _________\\\ / / _________\\
| |\/ 1204223 || | | |\/ 1204223 || | |\/ 1204223 ||| | |\/ 1204223 || | |\/ 1204223 || | | |\/ 1204223 || | |\/ 1204223 ||| | |\/ 1204223 || _________________________________
|atan|-----------|| | |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------|| | |atan|-----------|| |atan|-----------||| |atan|-----------|| 3 / _________
/ 1 \ | \ 602111 /| | / 1 \ | \ 602111 /| ___ | \ 602111 /|| ___ | \ 602111 /| / 1 \ | \ 602111 /| | / 1 \ | \ 602111 /| ___ | \ 602111 /|| ___ | \ 602111 /| 677376 2*\/ 1024190811 + 1701*I*\/ 1204223
-672 - 677376*re|----------------------------------------------------| + 336*cos|-----------------| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| + 336*\/ 3 *cos|-----------------|| - 336*\/ 3 *sin|-----------------| + -672 - 677376*re|----------------------------------------------------| + 336*cos|-----------------| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| - 336*\/ 3 *cos|-----------------|| + 336*\/ 3 *sin|-----------------| + -672 - ------------------------------------ - --------------------------------------
| _________________________________ / ___\| \ 3 / | | _________________________________ / ___\| \ 3 / \ 3 /| \ 3 / | _________________________________ / ___\| \ 3 / | | _________________________________ / ___\| \ 3 / \ 3 /| \ 3 / _________________________________ 3
|3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || | 3 / _________
|\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| | \/ 1024190811 + 1701*I*\/ 1204223
\ \ 2 2 // \ \ \ 2 2 // / \ \ 2 2 // \ \ \ 2 2 // /
$$\left(\left(-672 - 336 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + i \left(- 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)}\right)\right) + \left(-672 + 336 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + i \left(- 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)}\right)\right)\right) + \left(-672 - \frac{2 \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}{3} - \frac{677376}{\sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)$$
=
/ / _________\\ / / / _________\\ / / _________\\\ / / / _________\\ / / _________\\\
| |\/ 1204223 || _________________________________ | | |\/ 1204223 || | |\/ 1204223 ||| | | |\/ 1204223 || | |\/ 1204223 |||
|atan|-----------|| 3 / _________ | |atan|-----------|| |atan|-----------||| | |atan|-----------|| |atan|-----------|||
677376 / 1 \ / 1 \ | \ 602111 /| 2*\/ 1024190811 + 1701*I*\/ 1204223 | / 1 \ | \ 602111 /| ___ | \ 602111 /|| | / 1 \ | \ 602111 /| ___ | \ 602111 /||
-2016 - ------------------------------------ - 677376*re|----------------------------------------------------| - 677376*re|----------------------------------------------------| + 672*cos|-----------------| - -------------------------------------- + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| - 336*\/ 3 *cos|-----------------|| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| + 336*\/ 3 *cos|-----------------||
_________________________________ | _________________________________ / ___\| | _________________________________ / ___\| \ 3 / 3 | | _________________________________ / ___\| \ 3 / \ 3 /| | | _________________________________ / ___\| \ 3 / \ 3 /|
3 / _________ |3 / _________ | 1 I*\/ 3 || |3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || | | |3 / _________ | 1 I*\/ 3 || |
\/ 1024190811 + 1701*I*\/ 1204223 |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| |\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| | | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| |
\ \ 2 2 // \ \ 2 2 // \ \ \ 2 2 // / \ \ \ 2 2 // /
$$-2016 - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 672 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - \frac{2 \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}{3} + i \left(- 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)}\right) + i \left(- 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)}\right) - \frac{677376}{\sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}$$
producto
/ / / _________\\ / / / _________\\ / / _________\\\ / / _________\\\ / / / _________\\ / / / _________\\ / / _________\\\ / / _________\\\ | | |\/ 1204223 || | | |\/ 1204223 || | |\/ 1204223 ||| | |\/ 1204223 ||| | | |\/ 1204223 || | | |\/ 1204223 || | |\/ 1204223 ||| | |\/ 1204223 ||| / _________________________________\ | |atan|-----------|| | |atan|-----------|| |atan|-----------||| |atan|-----------||| | |atan|-----------|| | |atan|-----------|| |atan|-----------||| |atan|-----------||| | 3 / _________ | | / 1 \ | \ 602111 /| | / 1 \ | \ 602111 /| ___ | \ 602111 /|| ___ | \ 602111 /|| | / 1 \ | \ 602111 /| | / 1 \ | \ 602111 /| ___ | \ 602111 /|| ___ | \ 602111 /|| | 677376 2*\/ 1024190811 + 1701*I*\/ 1204223 | |-672 - 677376*re|----------------------------------------------------| + 336*cos|-----------------| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| + 336*\/ 3 *cos|-----------------|| - 336*\/ 3 *sin|-----------------||*|-672 - 677376*re|----------------------------------------------------| + 336*cos|-----------------| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| - 336*\/ 3 *cos|-----------------|| + 336*\/ 3 *sin|-----------------||*|-672 - ------------------------------------ - --------------------------------------| | | _________________________________ / ___\| \ 3 / | | _________________________________ / ___\| \ 3 / \ 3 /| \ 3 /| | | _________________________________ / ___\| \ 3 / | | _________________________________ / ___\| \ 3 / \ 3 /| \ 3 /| | _________________________________ 3 | | |3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || | | | |3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || | | | 3 / _________ | | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| | | | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| | | \ \/ 1024190811 + 1701*I*\/ 1204223 / \ \ \ 2 2 // \ \ \ 2 2 // / / \ \ \ 2 2 // \ \ \ 2 2 // / /
$$\left(-672 - 336 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + i \left(- 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)}\right)\right) \left(-672 + 336 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + i \left(- 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)}\right)\right) \left(-672 - \frac{2 \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}{3} - \frac{677376}{\sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)$$
=
/ / / _________\\ / / _________\\\ / / / _________\\ / / _________\\ \
| | |\/ 1204223 || | |\/ 1204223 ||| | | |\/ 1204223 || | |\/ 1204223 || |
/ _____________________________ / _____________________________\\ | |atan|-----------|| |atan|-----------||| | |atan|-----------|| |atan|-----------|| |
3 _____ | 3 / _________ | 3 / _________ || | ___ | \ 602111 /| 3 _____ / 1 \ | \ 602111 /|| | | \ 602111 /| ___ | \ 602111 /| 3 _____ / 1 \|
10752*\/ 147 *\112896 + \/ 37932993 + 63*I*\/ 1204223 *\336 + \/ 37932993 + 63*I*\/ 1204223 //*|-2 + \/ 3 *sin|-----------------| + 64*\/ 147 *re|-----------------------------------------| + cos|-----------------||*|2 - cos|-----------------| + \/ 3 *sin|-----------------| - 64*\/ 147 *re|-----------------------------------------||
| \ 3 / | ________________________| \ 3 /| | \ 3 / \ 3 / | ________________________||
| |/ ___\ 3 / _________ | | | |/ ___\ 3 / _________ ||
\ \\1 - I*\/ 3 /*\/ 602111 + I*\/ 1204223 / / \ \\1 + I*\/ 3 /*\/ 602111 + I*\/ 1204223 //
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
________________________
3 / _________
\/ 602111 + I*\/ 1204223
$$\frac{10752 \sqrt[3]{147} \left(112896 + \left(336 + \sqrt[3]{37932993 + 63 \sqrt{1204223} i}\right) \sqrt[3]{37932993 + 63 \sqrt{1204223} i}\right) \left(-2 + \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 64 \sqrt[3]{147} \operatorname{re}{\left(\frac{1}{\left(1 - \sqrt{3} i\right) \sqrt[3]{602111 + \sqrt{1204223} i}}\right)}\right) \left(- \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 64 \sqrt[3]{147} \operatorname{re}{\left(\frac{1}{\left(1 + \sqrt{3} i\right) \sqrt[3]{602111 + \sqrt{1204223} i}}\right)} + \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 2\right)}{\sqrt[3]{602111 + \sqrt{1204223} i}}$$
10752*147^(1/3)*(112896 + (37932993 + 63*i*sqrt(1204223))^(1/3)*(336 + (37932993 + 63*i*sqrt(1204223))^(1/3)))*(-2 + sqrt(3)*sin(atan(sqrt(1204223)/602111)/3) + 64*147^(1/3)*re(1/((1 - i*sqrt(3))*(602111 + i*sqrt(1204223))^(1/3))) + cos(atan(sqrt(1204223)/602111)/3))*(2 - cos(atan(sqrt(1204223)/602111)/3) + sqrt(3)*sin(atan(sqrt(1204223)/602111)/3) - 64*147^(1/3)*re(1/((1 + i*sqrt(3))*(602111 + i*sqrt(1204223))^(1/3))))/(602111 + i*sqrt(1204223))^(1/3)
Respuesta rápida
[src]
/ / _________\\ / / / _________\\ / / _________\\\ / / _________\\
| |\/ 1204223 || | | |\/ 1204223 || | |\/ 1204223 ||| | |\/ 1204223 ||
|atan|-----------|| | |atan|-----------|| |atan|-----------||| |atan|-----------||
/ 1 \ | \ 602111 /| | / 1 \ | \ 602111 /| ___ | \ 602111 /|| ___ | \ 602111 /|
x1 = -672 - 677376*re|----------------------------------------------------| + 336*cos|-----------------| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| + 336*\/ 3 *cos|-----------------|| - 336*\/ 3 *sin|-----------------|
| _________________________________ / ___\| \ 3 / | | _________________________________ / ___\| \ 3 / \ 3 /| \ 3 /
|3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || |
|\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - - -------|| |
\ \ 2 2 // \ \ \ 2 2 // /
$$x_{1} = -672 - 336 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + i \left(- 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)}\right)$$
/ / _________\\ / / / _________\\ / / _________\\\ / / _________\\
| |\/ 1204223 || | | |\/ 1204223 || | |\/ 1204223 ||| | |\/ 1204223 ||
|atan|-----------|| | |atan|-----------|| |atan|-----------||| |atan|-----------||
/ 1 \ | \ 602111 /| | / 1 \ | \ 602111 /| ___ | \ 602111 /|| ___ | \ 602111 /|
x2 = -672 - 677376*re|----------------------------------------------------| + 336*cos|-----------------| + I*|- 677376*im|----------------------------------------------------| + 336*sin|-----------------| - 336*\/ 3 *cos|-----------------|| + 336*\/ 3 *sin|-----------------|
| _________________________________ / ___\| \ 3 / | | _________________________________ / ___\| \ 3 / \ 3 /| \ 3 /
|3 / _________ | 1 I*\/ 3 || | |3 / _________ | 1 I*\/ 3 || |
|\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| | |\/ 1024190811 + 1701*I*\/ 1204223 *|- - + -------|| |
\ \ 2 2 // \ \ \ 2 2 // /
$$x_{2} = -672 + 336 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)} + i \left(- 336 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} + 336 \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{1204223}}{602111} \right)}}{3} \right)} - 677376 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}\right)}\right)$$
_________________________________
3 / _________
677376 2*\/ 1024190811 + 1701*I*\/ 1204223
x3 = -672 - ------------------------------------ - --------------------------------------
_________________________________ 3
3 / _________
\/ 1024190811 + 1701*I*\/ 1204223
$$x_{3} = -672 - \frac{2 \sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}{3} - \frac{677376}{\sqrt[3]{1024190811 + 1701 \sqrt{1204223} i}}$$
x3 = -672 - 2*(1024190811 + 1701*sqrt(1204223)*i)^(1/3)/3 - 677376/(1024190811 + 1701*sqrt(1204223)*i)^(1/3)
Respuesta numérica
[src]
x1 = -2015.99975198407
x2 = 0.706982827588875
x3 = -0.707230843522915
x3 = -0.707230843522915