Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(x*x-2*x-1)/2-(1+(x-1)/(x-1))/(x-1+(sqrt(x-1))^2-2)+(-1+x)*(x/2-1/2)/sqrt(x*x-2*x-1)
¿Cómo vas a descomponer esta sqrt(x*x-2*x-1)/2-(1+(x-1)/(x-1))/(x-1+(sqrt(x-1))^2-2)+(-1+x)*(x/2-1/2)/sqrt(x*x-2*x-1) expresión en fracciones?
Solución
Ha introducido
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x - 1 /x 1\
_______________ 1 + ----- (-1 + x)*|- - -|
\/ x*x - 2*x - 1 x - 1 \2 2/
----------------- - ---------------------- + -----------------
2 2 _______________
_______ \/ x*x - 2*x - 1
x - 1 + \/ x - 1 - 2
$$\frac{\left(x - 1\right) \left(\frac{x}{2} - \frac{1}{2}\right)}{\sqrt{\left(- 2 x + x x\right) - 1}} + \left(- \frac{1 + \frac{x - 1}{x - 1}}{\left(\left(\sqrt{x - 1}\right)^{2} + \left(x - 1\right)\right) - 2} + \frac{\sqrt{\left(- 2 x + x x\right) - 1}}{2}\right)$$
sqrt(x*x - 2*x - 1)/2 - (1 + (x - 1)/(x - 1))/(x - 1 + (sqrt(x - 1))^2 - 2) + ((-1 + x)*(x/2 - 1/2))/sqrt(x*x - 2*x - 1)
Simplificación general
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_______________
3 / 2 2
x - \/ -1 + x - 2*x - 4*x + 4*x
------------------------------------
_______________
/ 2
(-2 + x)*\/ -1 + x - 2*x
$$\frac{x^{3} - 4 x^{2} + 4 x - \sqrt{x^{2} - 2 x - 1}}{\left(x - 2\right) \sqrt{x^{2} - 2 x - 1}}$$
(x^3 - sqrt(-1 + x^2 - 2*x) - 4*x^2 + 4*x)/((-2 + x)*sqrt(-1 + x^2 - 2*x))
Respuesta numérica
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0.707106781186548*(-0.5 - x + 0.5*x^2)^0.5 - 2.0/(-3.0 + x + (-1.0 + x)^1.0) + 0.707106781186547*(-0.5 - x + 0.5*x^2)^(-0.5)*(-0.5 + 0.5*x)*(-1.0 + x)
0.707106781186548*(-0.5 - x + 0.5*x^2)^0.5 - 2.0/(-3.0 + x + (-1.0 + x)^1.0) + 0.707106781186547*(-0.5 - x + 0.5*x^2)^(-0.5)*(-0.5 + 0.5*x)*(-1.0 + x)
Combinatoria
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_______________
3 / 2 2
x - \/ -1 + x - 2*x - 4*x + 4*x
------------------------------------
_______________
/ 2
(-2 + x)*\/ -1 + x - 2*x
$$\frac{x^{3} - 4 x^{2} + 4 x - \sqrt{x^{2} - 2 x - 1}}{\left(x - 2\right) \sqrt{x^{2} - 2 x - 1}}$$
(x^3 - sqrt(-1 + x^2 - 2*x) - 4*x^2 + 4*x)/((-2 + x)*sqrt(-1 + x^2 - 2*x))
Potencias
[src]
_______________ / 1 x\
/ 2 (-1 + x)*|- - + -|
\/ -1 + x - 2*x 2 \ 2 2/
------------------ - -------- + ------------------
2 -4 + 2*x _______________
/ 2
\/ -1 + x - 2*x
$$\frac{\left(\frac{x}{2} - \frac{1}{2}\right) \left(x - 1\right)}{\sqrt{x^{2} - 2 x - 1}} + \frac{\sqrt{x^{2} - 2 x - 1}}{2} - \frac{2}{2 x - 4}$$
sqrt(-1 + x^2 - 2*x)/2 - 2/(-4 + 2*x) + (-1 + x)*(-1/2 + x/2)/sqrt(-1 + x^2 - 2*x)
Parte trigonométrica
[src]
_______________ / 1 x\
/ 2 (-1 + x)*|- - + -|
\/ -1 + x - 2*x 2 \ 2 2/
------------------ - -------- + ------------------
2 -4 + 2*x _______________
/ 2
\/ -1 + x - 2*x
$$\frac{\left(\frac{x}{2} - \frac{1}{2}\right) \left(x - 1\right)}{\sqrt{x^{2} - 2 x - 1}} + \frac{\sqrt{x^{2} - 2 x - 1}}{2} - \frac{2}{2 x - 4}$$
sqrt(-1 + x^2 - 2*x)/2 - 2/(-4 + 2*x) + (-1 + x)*(-1/2 + x/2)/sqrt(-1 + x^2 - 2*x)
Denominador común
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_______________
3 / 2 2
x - \/ -1 + x - 2*x - 4*x + 4*x
---------------------------------------------
_______________ _______________
/ 2 / 2
- 2*\/ -1 + x - 2*x + x*\/ -1 + x - 2*x
$$\frac{x^{3} - 4 x^{2} + 4 x - \sqrt{x^{2} - 2 x - 1}}{x \sqrt{x^{2} - 2 x - 1} - 2 \sqrt{x^{2} - 2 x - 1}}$$
(x^3 - sqrt(-1 + x^2 - 2*x) - 4*x^2 + 4*x)/(-2*sqrt(-1 + x^2 - 2*x) + x*sqrt(-1 + x^2 - 2*x))
Denominador racional
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_______________ _______________ _______________ 3/2 3/2 _______________ _______________ _______________
2 3 / 2 2 / 2 2 / 2 2 / 2 \ / 2 \ / 2 2 2 / 2 2 / 2
16 - 48*x + 16*x + 16*x + 16*\/ -1 + x - 2*x - 16*x *\/ -1 + x - 2*x - 16*(-1 + x) *\/ -1 + x - 2*x - 8*x *\-1 + x - 2*x/ + 24*x*\-1 + x - 2*x/ + 32*x*\/ -1 + x - 2*x - 8*x *(-1 + x) *\/ -1 + x - 2*x + 24*x*(-1 + x) *\/ -1 + x - 2*x
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2 \
8*(-1 + x)*(-4 + 2*x)*\1 - x + 2*x/
$$\frac{16 x^{3} - 8 x^{2} \left(x - 1\right)^{2} \sqrt{x^{2} - 2 x - 1} - 8 x^{2} \left(x^{2} - 2 x - 1\right)^{\frac{3}{2}} - 16 x^{2} \sqrt{x^{2} - 2 x - 1} - 48 x^{2} + 24 x \left(x - 1\right)^{2} \sqrt{x^{2} - 2 x - 1} + 24 x \left(x^{2} - 2 x - 1\right)^{\frac{3}{2}} + 32 x \sqrt{x^{2} - 2 x - 1} + 16 x - 16 \left(x - 1\right)^{2} \sqrt{x^{2} - 2 x - 1} + 16 \sqrt{x^{2} - 2 x - 1} + 16}{8 \left(x - 1\right) \left(2 x - 4\right) \left(- x^{2} + 2 x + 1\right)}$$
(16 - 48*x^2 + 16*x + 16*x^3 + 16*sqrt(-1 + x^2 - 2*x) - 16*x^2*sqrt(-1 + x^2 - 2*x) - 16*(-1 + x)^2*sqrt(-1 + x^2 - 2*x) - 8*x^2*(-1 + x^2 - 2*x)^(3/2) + 24*x*(-1 + x^2 - 2*x)^(3/2) + 32*x*sqrt(-1 + x^2 - 2*x) - 8*x^2*(-1 + x)^2*sqrt(-1 + x^2 - 2*x) + 24*x*(-1 + x)^2*sqrt(-1 + x^2 - 2*x))/(8*(-1 + x)*(-4 + 2*x)*(1 - x^2 + 2*x))
Compilar la expresión
[src]
_______________ / 1 x\
/ 2 (-1 + x)*|- - + -|
\/ -1 + x - 2*x 2 \ 2 2/
------------------ - -------- + ------------------
2 -4 + 2*x _______________
/ 2
\/ -1 + x - 2*x
$$\frac{\left(\frac{x}{2} - \frac{1}{2}\right) \left(x - 1\right)}{\sqrt{x^{2} - 2 x - 1}} + \frac{\sqrt{x^{2} - 2 x - 1}}{2} - \frac{2}{2 x - 4}$$
sqrt(-1 + x^2 - 2*x)/2 - 2/(-4 + 2*x) + (-1 + x)*(-1/2 + x/2)/sqrt(-1 + x^2 - 2*x)
Unión de expresiones racionales
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2 _________________ / _________________ \
(-1 + x) *(-2 + x) + \/ -1 + x*(-2 + x) *\-2 + \/ -1 + x*(-2 + x) *(-2 + x)/
----------------------------------------------------------------------------
_________________
2*\/ -1 + x*(-2 + x) *(-2 + x)
$$\frac{\left(x - 2\right) \left(x - 1\right)^{2} + \sqrt{x \left(x - 2\right) - 1} \left(\left(x - 2\right) \sqrt{x \left(x - 2\right) - 1} - 2\right)}{2 \left(x - 2\right) \sqrt{x \left(x - 2\right) - 1}}$$
((-1 + x)^2*(-2 + x) + sqrt(-1 + x*(-2 + x))*(-2 + sqrt(-1 + x*(-2 + x))*(-2 + x)))/(2*sqrt(-1 + x*(-2 + x))*(-2 + x))