absolute((x-sqrt(2))*(x-sqrt(3)))=0,025 la ecuación
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Solución
Solución detallada
Para cada expresión dentro del módulo en la ecuación
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.
1.
$$\left(x - \sqrt{2}\right) \left(x - \sqrt{3}\right) \geq 0$$
o
$$\left(x \leq - \frac{\sqrt{5 - 2 \sqrt{6}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \wedge -\infty < x\right) \vee \left(\frac{\sqrt{5 - 2 \sqrt{6}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \leq x \wedge x < \infty\right)$$
obtenemos la ecuación
$$\left(x - \sqrt{2}\right) \left(x - \sqrt{3}\right) - \frac{1}{40} = 0$$
simplificamos, obtenemos
$$\left(x - \sqrt{2}\right) \left(x - \sqrt{3}\right) - \frac{1}{40} = 0$$
la resolución en este intervalo:
$$x_{1} = - \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
$$x_{2} = \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
2.
$$\left(x - \sqrt{2}\right) \left(x - \sqrt{3}\right) < 0$$
o
$$x < \frac{\sqrt{5 - 2 \sqrt{6}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \wedge - \frac{\sqrt{5 - 2 \sqrt{6}}}{2} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} < x$$
obtenemos la ecuación
$$- \left(x - \sqrt{2}\right) \left(x - \sqrt{3}\right) - \frac{1}{40} = 0$$
simplificamos, obtenemos
$$- \left(x - \sqrt{2}\right) \left(x - \sqrt{3}\right) - \frac{1}{40} = 0$$
la resolución en este intervalo:
$$x_{3} = - \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
$$x_{4} = \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
Entonces la respuesta definitiva es:
$$x_{1} = - \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
$$x_{2} = \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
$$x_{3} = - \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
$$x_{4} = \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
Suma y producto de raíces
[src]
_________________ _________________ _________________ _________________
___ ___ / ___ ___ ___ / ___ ___ ___ / ___ ___ ___ / ___
\/ 2 \/ 3 \/ 490 - 200*\/ 6 \/ 2 \/ 3 \/ 490 - 200*\/ 6 \/ 2 \/ 3 \/ 510 - 200*\/ 6 \/ 2 \/ 3 \/ 510 - 200*\/ 6
----- + ----- - -------------------- + ----- + ----- + -------------------- + ----- + ----- - -------------------- + ----- + ----- + --------------------
2 2 20 2 2 20 2 2 20 2 2 20
$$\left(\frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right) + \left(\left(- \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right) + \left(\left(- \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right)\right)\right)$$
$$2 \sqrt{2} + 2 \sqrt{3}$$
/ _________________\ / _________________\ / _________________\ / _________________\
| ___ ___ / ___ | | ___ ___ / ___ | | ___ ___ / ___ | | ___ ___ / ___ |
|\/ 2 \/ 3 \/ 490 - 200*\/ 6 | |\/ 2 \/ 3 \/ 490 - 200*\/ 6 | |\/ 2 \/ 3 \/ 510 - 200*\/ 6 | |\/ 2 \/ 3 \/ 510 - 200*\/ 6 |
|----- + ----- - --------------------|*|----- + ----- + --------------------|*|----- + ----- - --------------------|*|----- + ----- + --------------------|
\ 2 2 20 / \ 2 2 20 / \ 2 2 20 / \ 2 2 20 /
$$\left(- \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right) \left(- \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\right)$$
$$\frac{9599}{1600}$$
_________________
___ ___ / ___
\/ 2 \/ 3 \/ 490 - 200*\/ 6
x1 = ----- + ----- - --------------------
2 2 20
$$x_{1} = - \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
_________________
___ ___ / ___
\/ 2 \/ 3 \/ 490 - 200*\/ 6
x2 = ----- + ----- + --------------------
2 2 20
$$x_{2} = \frac{\sqrt{490 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
_________________
___ ___ / ___
\/ 2 \/ 3 \/ 510 - 200*\/ 6
x3 = ----- + ----- - --------------------
2 2 20
$$x_{3} = - \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
_________________
___ ___ / ___
\/ 2 \/ 3 \/ 510 - 200*\/ 6
x4 = ----- + ----- + --------------------
2 2 20
$$x_{4} = \frac{\sqrt{510 - 200 \sqrt{6}}}{20} + \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}$$
x4 = sqrt(510 - 200*sqrt(6))/20 + sqrt(2)/2 + sqrt(3)/2