Answered

At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

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.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.