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- sqrt(x^2-36)+sqrt(x^2-64)=28/(x-8)
sqrt(3)*x^3+3*x^2-3*sqrt(3)*x-1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
___ 3 2 ___ \/ 3 *x + 3*x - 3*\/ 3 *x - 1 = 0
$$\left(- 3 \sqrt{3} x + \left(\sqrt{3} x^{3} + 3 x^{2}\right)\right) - 1 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 3 \sqrt{3} x + \left(\sqrt{3} x^{3} + 3 x^{2}\right)\right) - 1 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$\frac{\sqrt{3} \left(\sqrt{3} x^{3} + 3 x^{2} - 3 \sqrt{3} x - 1\right)}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \sqrt{3}$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = - \frac{\sqrt{3}}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \sqrt{3}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = - \frac{\sqrt{3}}{3}$$
$$\left(- 3 \sqrt{3} x + \left(\sqrt{3} x^{3} + 3 x^{2}\right)\right) - 1 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$\frac{\sqrt{3} \left(\sqrt{3} x^{3} + 3 x^{2} - 3 \sqrt{3} x - 1\right)}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = \sqrt{3}$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = - \frac{\sqrt{3}}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \sqrt{3}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = - \frac{\sqrt{3}}{3}$$
Respuesta rápida
[src]
/ ___ /pi\ \ ___ /pi\
___ | \/ 3 *sin|--| | \/ 3 *cos|--|
/pi\ / 1 \ \/ 3 | / 1 \ \9 / /pi\| \9 /
x1 = - sin|--| - 4*re|------------------------------------| - ----- + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + -------------
\9 / |/ ___\ _________________| 3 | |/ ___\ _________________| 3 \9 /| 3
|| 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | |
||- - - -------|*\/ 12*\/ 3 + 36*I | | ||- - - -------|*\/ 12*\/ 3 + 36*I | |
\\ 2 2 / / \ \\ 2 2 / / /
$$x_{1} = - \frac{\sqrt{3}}{3} - \sin{\left(\frac{\pi}{9} \right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right)$$
/ ___ /pi\\ ___ /pi\
___ | \/ 3 *sin|--|| \/ 3 *cos|--|
/ 1 \ \/ 3 | /pi\ / 1 \ \9 /| \9 / /pi\
x2 = - 4*re|------------------------------------| - ----- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + ------------- + sin|--|
|/ ___\ _________________| 3 | \9 / |/ ___\ _________________| 3 | 3 \9 /
|| 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | |
||- - + -------|*\/ 12*\/ 3 + 36*I | | ||- - + -------|*\/ 12*\/ 3 + 36*I | |
\\ 2 2 / / \ \\ 2 2 / / /
$$x_{2} = - \frac{\sqrt{3}}{3} + \sin{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right)$$
_________________
___ 3 / ___
4 \/ 3 \/ 12*\/ 3 + 36*I
x3 = - -------------------- - ----- - --------------------
_________________ 3 3
3 / ___
\/ 12*\/ 3 + 36*I
$$x_{3} = - \frac{\sqrt{3}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}}$$
x3 = -sqrt(3)/3 - (12*sqrt(3) + 36*i)^(1/3)/3 - 4/(12*sqrt(3) + 36*i)^(1/3)
Suma y producto de raíces
[src]
suma
/ ___ /pi\ \ ___ /pi\ / ___ /pi\\ ___ /pi\ _________________
___ | \/ 3 *sin|--| | \/ 3 *cos|--| ___ | \/ 3 *sin|--|| \/ 3 *cos|--| ___ 3 / ___
/pi\ / 1 \ \/ 3 | / 1 \ \9 / /pi\| \9 / / 1 \ \/ 3 | /pi\ / 1 \ \9 /| \9 / /pi\ 4 \/ 3 \/ 12*\/ 3 + 36*I
- sin|--| - 4*re|------------------------------------| - ----- + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + ------------- + - 4*re|------------------------------------| - ----- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + ------------- + sin|--| + - -------------------- - ----- - --------------------
\9 / |/ ___\ _________________| 3 | |/ ___\ _________________| 3 \9 /| 3 |/ ___\ _________________| 3 | \9 / |/ ___\ _________________| 3 | 3 \9 / _________________ 3 3
|| 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | | 3 / ___
||- - - -------|*\/ 12*\/ 3 + 36*I | | ||- - - -------|*\/ 12*\/ 3 + 36*I | | ||- - + -------|*\/ 12*\/ 3 + 36*I | | ||- - + -------|*\/ 12*\/ 3 + 36*I | | \/ 12*\/ 3 + 36*I
\\ 2 2 / / \ \\ 2 2 / / / \\ 2 2 / / \ \\ 2 2 / / /
$$\left(\left(- \frac{\sqrt{3}}{3} + \sin{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right)\right) + \left(- \frac{\sqrt{3}}{3} - \sin{\left(\frac{\pi}{9} \right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right)\right)\right) + \left(- \frac{\sqrt{3}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}}\right)$$
=
_________________ / ___ /pi\\ / ___ /pi\ \ ___ /pi\
3 / ___ | \/ 3 *sin|--|| | \/ 3 *sin|--| | 2*\/ 3 *cos|--|
___ 4 / 1 \ / 1 \ \/ 12*\/ 3 + 36*I | /pi\ / 1 \ \9 /| | / 1 \ \9 / /pi\| \9 /
- \/ 3 - -------------------- - 4*re|------------------------------------| - 4*re|------------------------------------| - -------------------- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + ---------------
_________________ |/ ___\ _________________| |/ ___\ _________________| 3 | \9 / |/ ___\ _________________| 3 | | |/ ___\ _________________| 3 \9 /| 3
3 / ___ || 1 I*\/ 3 | 3 / ___ | || 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | | | || 1 I*\/ 3 | 3 / ___ | |
\/ 12*\/ 3 + 36*I ||- - + -------|*\/ 12*\/ 3 + 36*I | ||- - - -------|*\/ 12*\/ 3 + 36*I | | ||- - + -------|*\/ 12*\/ 3 + 36*I | | | ||- - - -------|*\/ 12*\/ 3 + 36*I | |
\\ 2 2 / / \\ 2 2 / / \ \\ 2 2 / / / \ \\ 2 2 / / /
$$- \sqrt{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{2 \sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right) + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right) - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}}$$
producto
/ / ___ /pi\ \ ___ /pi\\ / / ___ /pi\\ ___ /pi\ \ / _________________\ | ___ | \/ 3 *sin|--| | \/ 3 *cos|--|| | ___ | \/ 3 *sin|--|| \/ 3 *cos|--| | | ___ 3 / ___ | | /pi\ / 1 \ \/ 3 | / 1 \ \9 / /pi\| \9 /| | / 1 \ \/ 3 | /pi\ / 1 \ \9 /| \9 / /pi\| | 4 \/ 3 \/ 12*\/ 3 + 36*I | |- sin|--| - 4*re|------------------------------------| - ----- + I*|- 4*im|------------------------------------| + ------------- + cos|--|| + -------------|*|- 4*re|------------------------------------| - ----- + I*|- cos|--| - 4*im|------------------------------------| + -------------| + ------------- + sin|--||*|- -------------------- - ----- - --------------------| | \9 / |/ ___\ _________________| 3 | |/ ___\ _________________| 3 \9 /| 3 | | |/ ___\ _________________| 3 | \9 / |/ ___\ _________________| 3 | 3 \9 /| | _________________ 3 3 | | || 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | | | | || 1 I*\/ 3 | 3 / ___ | | || 1 I*\/ 3 | 3 / ___ | | | | 3 / ___ | | ||- - - -------|*\/ 12*\/ 3 + 36*I | | ||- - - -------|*\/ 12*\/ 3 + 36*I | | | | ||- - + -------|*\/ 12*\/ 3 + 36*I | | ||- - + -------|*\/ 12*\/ 3 + 36*I | | | \ \/ 12*\/ 3 + 36*I / \ \\ 2 2 / / \ \\ 2 2 / / / / \ \\ 2 2 / / \ \\ 2 2 / / / /
$$\left(- \frac{\sqrt{3}}{3} - \sin{\left(\frac{\pi}{9} \right)} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} + i \left(- 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} + \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(- \frac{\sqrt{3}}{3} + \sin{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)} + i \left(- \cos{\left(\frac{\pi}{9} \right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{3} - 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{12 \sqrt{3} + 36 i}}\right)}\right)\right) \left(- \frac{\sqrt{3}}{3} - \frac{\sqrt[3]{12 \sqrt{3} + 36 i}}{3} - \frac{4}{\sqrt[3]{12 \sqrt{3} + 36 i}}\right)$$
=
/ _________________ / _________________\\
3 ____ | 3 / ___ | ___ 3 / ___ || / /2*pi\ /pi\\
\/ 18 *\12 + \/ 12*\/ 3 + 36*I *\\/ 3 + \/ 12*\/ 3 + 36*I //*|9 - 24*cos|----| + 12*cos|--||
\ \ 9 / \9 //
-------------------------------------------------------------------------------------------------
_____________
3 / ___
162*\/ \/ 3 + 3*I
$$\frac{\sqrt[3]{18} \left(12 + \left(\sqrt{3} + \sqrt[3]{12 \sqrt{3} + 36 i}\right) \sqrt[3]{12 \sqrt{3} + 36 i}\right) \left(- 24 \cos{\left(\frac{2 \pi}{9} \right)} + 9 + 12 \cos{\left(\frac{\pi}{9} \right)}\right)}{162 \sqrt[3]{\sqrt{3} + 3 i}}$$
18^(1/3)*(12 + (12*sqrt(3) + 36*i)^(1/3)*(sqrt(3) + (12*sqrt(3) + 36*i)^(1/3)))*(9 - 24*cos(2*pi/9) + 12*cos(pi/9))/(162*(sqrt(3) + 3*i)^(1/3))
Respuesta numérica
[src]
x1 = -2.74747741945462
x2 = -0.176326980708465
x3 = 1.19175359259421
x3 = 1.19175359259421