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Calculate the value of [tex]\sqrt{x+y+2} + \sqrt{x+y} - 3[/tex], given that [tex]x^2 + y^2 = 25[/tex] and [tex]xy = 12[/tex].

Sagot :

GinnyAnswer
Primero, tenemos las siguientes dos ecuaciones:

[tex]\[ x^2 + y^2 = 25 \][/tex]
[tex]\[ xy = 12 \][/tex]

Nuestro objetivo es encontrar los valores de [tex]\( x \)[/tex] y [tex]\( y \)[/tex] que satisfacen ambas ecuaciones y luego usar estos valores para calcular la expresión:

[tex]\[ 1 = \sqrt{x + y + 2} + \sqrt{x + y} - 3 \][/tex]

A partir de las ecuaciones dadas, resolvemos el sistema para hallar [tex]\( x \)[/tex] y [tex]\( y \)[/tex]. Esto nos da los pares de soluciones:

[tex]\[ (x, y) = (-4, -3), (-3, -4), (3, 4), (4, 3) \][/tex]

Ahora evaluamos la expresión [tex]\( \sqrt{x + y + 2} + \sqrt{x + y} - 3 \)[/tex] para cada par de soluciones:

1. Para [tex]\( (x, y) = (-4, -3) \)[/tex]:
[tex]\[ x + y = -4 - 3 = -7 \][/tex]
[tex]\[ \sqrt{x + y + 2} = \sqrt{-7 + 2} = \sqrt{-5} \][/tex]
[tex]\[ \sqrt{x + y} = \sqrt{-7} \][/tex]
[tex]\[ \sqrt{-5} + \sqrt{-7} - 3 = \sqrt{5}i + \sqrt{7}i - 3 \][/tex]

2. Para [tex]\( (x, y) = (-3, -4) \)[/tex]:
[tex]\[ x + y = -3 - 4 = -7 \][/tex]
[tex]\[ \sqrt{x + y + 2} = \sqrt{-7 + 2} = \sqrt{-5} \][/tex]
[tex]\[ \sqrt{x + y} = \sqrt{-7} \][/tex]
[tex]\[ \sqrt{-5} + \sqrt{-7} - 3 = \sqrt{5}i + \sqrt{7}i - 3 \][/tex]

3. Para [tex]\( (x, y) = (3, 4) \)[/tex]:
[tex]\[ x + y = 3 + 4 = 7 \][/tex]
[tex]\[ \sqrt{x + y + 2} = \sqrt{7 + 2} = \sqrt{9} = 3 \][/tex]
[tex]\[ \sqrt{x + y} = \sqrt{7} \][/tex]
[tex]\[ 3 + \sqrt{7} - 3 = \sqrt{7} \][/tex]

4. Para [tex]\( (x, y) = (4, 3) \)[/tex]:
[tex]\[ x + y = 4 + 3 = 7 \][/tex]
[tex]\[ \sqrt{x + y + 2} = \sqrt{7 + 2} = \sqrt{9} = 3 \][/tex]
[tex]\[ \sqrt{x + y} = \sqrt{7} \][/tex]
[tex]\[ 3 + \sqrt{7} - 3 = \sqrt{7} \][/tex]

De todas las posibles soluciones, observamos que:

Para [tex]\( (x, y) = (3, 4) \)[/tex] y [tex]\( (x, y) = (4, 3) \)[/tex]:

[tex]\[ 1 = \sqrt{7} \][/tex]

Para [tex]\( (x, y) = (-4, -3) \)[/tex] y [tex]\( (x, y) = (-3, -4) \)[/tex]:

[tex]\[ 1 = \sqrt{5}i + \sqrt{7}i - 3 \][/tex]

Conclusión:

La solución calculada para la expresión dada, cuando [tex]\( (x, y) = (3, 4) \)[/tex] y [tex]\( (x, y) = (4, 3) \)[/tex], es:

[tex]\[ \boxed{\sqrt{7}} \][/tex]

En los otros casos, la parte real no se mantiene igual a 1 dado que [tex]\( \sqrt{5}i + \sqrt{7}i - 3 \)[/tex] implica números complejos.
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