Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(x-2*sqrt(x+3)+4)/((1-sqrt(x+3))*(sqrt(x)-sqrt(x+3)))-(1/(sqrt(x)+sqrt(x-3)))
¿Cómo vas a descomponer esta sqrt(x-2*sqrt(x+3)+4)/((1-sqrt(x+3))*(sqrt(x)-sqrt(x+3)))-(1/(sqrt(x)+sqrt(x-3))) expresión en fracciones?
Solución
Ha introducido
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_____________________
/ _______
\/ x - 2*\/ x + 3 + 4 1
----------------------------------- - -----------------
/ _______\ / ___ _______\ ___ _______
\1 - \/ x + 3 /*\\/ x - \/ x + 3 / \/ x + \/ x - 3
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\sqrt{\left(x - 2 \sqrt{x + 3}\right) + 4}}{\left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right)}$$
sqrt(x - 2*sqrt(x + 3) + 4)/(((1 - sqrt(x + 3))*(sqrt(x) - sqrt(x + 3)))) - 1/(sqrt(x) + sqrt(x - 3))
Simplificación general
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/ _____________________ \
| / _______ / ___ ________\ / _______\ / ___ _______\|
-\\/ 4 + x - 2*\/ 3 + x *\\/ x + \/ -3 + x / + \-1 + \/ 3 + x /*\\/ x - \/ 3 + x //
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/ _______\ / ___ ________\ / ___ _______\
\-1 + \/ 3 + x /*\\/ x + \/ -3 + x /*\\/ x - \/ 3 + x /
$$- \frac{\left(\sqrt{x} + \sqrt{x - 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4} + \left(\sqrt{x} - \sqrt{x + 3}\right) \left(\sqrt{x + 3} - 1\right)}{\left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right) \left(\sqrt{x + 3} - 1\right)}$$
-(sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x)) + (-1 + sqrt(3 + x))*(sqrt(x) - sqrt(3 + x)))/((-1 + sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x))*(sqrt(x) - sqrt(3 + x)))
Respuesta numérica
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-1/(x^0.5 + 1.73205080756888*(-1 + 0.333333333333333*x)^0.5) + 2.0*(1 + 0.25*x - 0.866025403784439*(1 + 0.333333333333333*x)^0.5)^0.5/((1.0 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5)*(x^0.5 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5))
-1/(x^0.5 + 1.73205080756888*(-1 + 0.333333333333333*x)^0.5) + 2.0*(1 + 0.25*x - 0.866025403784439*(1 + 0.333333333333333*x)^0.5)^0.5/((1.0 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5)*(x^0.5 - 1.73205080756888*(1 + 0.333333333333333*x)^0.5))
Compilar la expresión
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_____________________
/ _______
1 \/ 4 + x - 2*\/ 3 + x
- ------------------ + -----------------------------------
___ ________ / _______\ / ___ _______\
\/ x + \/ -3 + x \1 - \/ 3 + x /*\\/ x - \/ 3 + x /
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\sqrt{x - 2 \sqrt{x + 3} + 4}}{\left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right)}$$
-1/(sqrt(x) + sqrt(-3 + x)) + sqrt(4 + x - 2*sqrt(3 + x))/((1 - sqrt(3 + x))*(sqrt(x) - sqrt(3 + x)))
Potencias
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_____________________
/ _______
1 \/ 4 + x - 2*\/ 3 + x
- ------------------ + -----------------------------------
___ ________ / _______\ / ___ _______\
\/ x + \/ -3 + x \1 - \/ 3 + x /*\\/ x - \/ 3 + x /
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\sqrt{x - 2 \sqrt{x + 3} + 4}}{\left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right)}$$
-1/(sqrt(x) + sqrt(-3 + x)) + sqrt(4 + x - 2*sqrt(3 + x))/((1 - sqrt(3 + x))*(sqrt(x) - sqrt(3 + x)))
Unión de expresiones racionales
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_____________________
/ _______ / ___ ________\ / _______\ / ___ _______\
\/ 4 + x - 2*\/ 3 + x *\\/ x + \/ -3 + x / - \1 - \/ 3 + x /*\\/ x - \/ 3 + x /
-----------------------------------------------------------------------------------
/ _______\ / ___ ________\ / ___ _______\
\1 - \/ 3 + x /*\\/ x + \/ -3 + x /*\\/ x - \/ 3 + x /
$$\frac{- \left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right) + \left(\sqrt{x} + \sqrt{x - 3}\right) \sqrt{x - 2 \sqrt{x + 3} + 4}}{\left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right)}$$
(sqrt(4 + x - 2*sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x)) - (1 - sqrt(3 + x))*(sqrt(x) - sqrt(3 + x)))/((1 - sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x))*(sqrt(x) - sqrt(3 + x)))
Denominador común
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_____________________ _____________________
___ _______ ___ _______ ___ / _______ ________ / _______
3 + x + \/ x - \/ 3 + x - \/ x *\/ 3 + x - \/ x *\/ 4 + x - 2*\/ 3 + x - \/ -3 + x *\/ 4 + x - 2*\/ 3 + x
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3/2 ___ ________ _______ ___ _______ ________ _______ ________ ___ ________ ___ ________ _______
-x - x - 3*\/ x - 3*\/ -3 + x + x*\/ 3 + x + \/ x *\/ 3 + x + \/ -3 + x *\/ 3 + x - x*\/ -3 + x - \/ x *\/ -3 + x + \/ x *\/ -3 + x *\/ 3 + x
$$\frac{- \sqrt{x} \sqrt{x + 3} - \sqrt{x} \sqrt{x - 2 \sqrt{x + 3} + 4} + \sqrt{x} + x - \sqrt{x - 3} \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x + 3} + 3}{- x^{\frac{3}{2}} + \sqrt{x} \sqrt{x - 3} \sqrt{x + 3} - \sqrt{x} \sqrt{x - 3} + \sqrt{x} \sqrt{x + 3} - 3 \sqrt{x} - x \sqrt{x - 3} + x \sqrt{x + 3} - x + \sqrt{x - 3} \sqrt{x + 3} - 3 \sqrt{x - 3}}$$
(3 + x + sqrt(x) - sqrt(3 + x) - sqrt(x)*sqrt(3 + x) - sqrt(x)*sqrt(4 + x - 2*sqrt(3 + x)) - sqrt(-3 + x)*sqrt(4 + x - 2*sqrt(3 + x)))/(-x - x^(3/2) - 3*sqrt(x) - 3*sqrt(-3 + x) + x*sqrt(3 + x) + sqrt(x)*sqrt(3 + x) + sqrt(-3 + x)*sqrt(3 + x) - x*sqrt(-3 + x) - sqrt(x)*sqrt(-3 + x) + sqrt(x)*sqrt(-3 + x)*sqrt(3 + x))
Combinatoria
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/ _____________________ _____________________\
| ___ _______ ___ _______ ___ / _______ ________ / _______ |
-\3 + x + \/ x - \/ 3 + x - \/ x *\/ 3 + x - \/ x *\/ 4 + x - 2*\/ 3 + x - \/ -3 + x *\/ 4 + x - 2*\/ 3 + x /
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/ _______\ / ___ ________\ / _______ ___\
\-1 + \/ 3 + x /*\\/ x + \/ -3 + x /*\\/ 3 + x - \/ x /
$$- \frac{- \sqrt{x} \sqrt{x + 3} - \sqrt{x} \sqrt{x - 2 \sqrt{x + 3} + 4} + \sqrt{x} + x - \sqrt{x - 3} \sqrt{x - 2 \sqrt{x + 3} + 4} - \sqrt{x + 3} + 3}{\left(- \sqrt{x} + \sqrt{x + 3}\right) \left(\sqrt{x} + \sqrt{x - 3}\right) \left(\sqrt{x + 3} - 1\right)}$$
-(3 + x + sqrt(x) - sqrt(3 + x) - sqrt(x)*sqrt(3 + x) - sqrt(x)*sqrt(4 + x - 2*sqrt(3 + x)) - sqrt(-3 + x)*sqrt(4 + x - 2*sqrt(3 + x)))/((-1 + sqrt(3 + x))*(sqrt(x) + sqrt(-3 + x))*(sqrt(3 + x) - sqrt(x)))
Parte trigonométrica
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_____________________
/ _______
1 \/ 4 + x - 2*\/ 3 + x
- ------------------ + -----------------------------------
___ ________ / _______\ / ___ _______\
\/ x + \/ -3 + x \1 - \/ 3 + x /*\\/ x - \/ 3 + x /
$$- \frac{1}{\sqrt{x} + \sqrt{x - 3}} + \frac{\sqrt{x - 2 \sqrt{x + 3} + 4}}{\left(1 - \sqrt{x + 3}\right) \left(\sqrt{x} - \sqrt{x + 3}\right)}$$
-1/(sqrt(x) + sqrt(-3 + x)) + sqrt(4 + x - 2*sqrt(3 + x))/((1 - sqrt(3 + x))*(sqrt(x) - sqrt(3 + x)))
Denominador racional
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_____________________ _____________________ _____________________ _____________________ _____________________
3/2 ___ ________ / _______ ________ / _______ ___ / _______ _______ / _______ ___ _______ / _______
- x - 2*\/ x + 2*\/ -3 + x + 3*\/ 4 + x - 2*\/ 3 + x + x*\/ -3 + x + x*\/ 4 + x - 2*\/ 3 + x + \/ x *\/ 4 + x - 2*\/ 3 + x + \/ 3 + x *\/ 4 + x - 2*\/ 3 + x + \/ x *\/ 3 + x *\/ 4 + x - 2*\/ 3 + x
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6 + 3*x
$$\frac{- x^{\frac{3}{2}} + \sqrt{x} \sqrt{x + 3} \sqrt{x - 2 \sqrt{x + 3} + 4} + \sqrt{x} \sqrt{x - 2 \sqrt{x + 3} + 4} - 2 \sqrt{x} + x \sqrt{x - 3} + x \sqrt{x - 2 \sqrt{x + 3} + 4} + 2 \sqrt{x - 3} + \sqrt{x + 3} \sqrt{x - 2 \sqrt{x + 3} + 4} + 3 \sqrt{x - 2 \sqrt{x + 3} + 4}}{3 x + 6}$$
(-x^(3/2) - 2*sqrt(x) + 2*sqrt(-3 + x) + 3*sqrt(4 + x - 2*sqrt(3 + x)) + x*sqrt(-3 + x) + x*sqrt(4 + x - 2*sqrt(3 + x)) + sqrt(x)*sqrt(4 + x - 2*sqrt(3 + x)) + sqrt(3 + x)*sqrt(4 + x - 2*sqrt(3 + x)) + sqrt(x)*sqrt(3 + x)*sqrt(4 + x - 2*sqrt(3 + x)))/(6 + 3*x)