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Sagot :
Claro, vamos a resolver la expresión paso a paso.
Primero, sabemos que [tex]\( r = \sqrt{2} + 1 \)[/tex]. Queremos encontrar el valor de la expresión [tex]\( r^2 - 2r - 2 \)[/tex].
### Paso 1: Calcula [tex]\( r^2 \)[/tex]
Dado que [tex]\( r = \sqrt{2} + 1 \)[/tex]:
[tex]\[ r^2 = (\sqrt{2} + 1)^2 \][/tex]
Usamos el cuadrado de un binomio:
[tex]\[ r^2 = (\sqrt{2})^2 + 2 \cdot (\sqrt{2}) \cdot 1 + 1^2 \][/tex]
[tex]\[ r^2 = 2 + 2\sqrt{2} + 1 \][/tex]
[tex]\[ r^2 = 3 + 2\sqrt{2} \][/tex]
### Paso 2: Calcula [tex]\(-2r\)[/tex]
Dado que [tex]\( r = \sqrt{2} + 1 \)[/tex]:
[tex]\[ -2r = -2(\sqrt{2} + 1) \][/tex]
[tex]\[ -2r = -2\sqrt{2} - 2 \][/tex]
### Paso 3: Sustituye los valores calculados en la expresión original
Ahora sustituimos [tex]\( r^2 \)[/tex] y [tex]\(-2r\)[/tex] en la expresión [tex]\( r^2 - 2r - 2 \)[/tex]:
[tex]\[ r^2 - 2r - 2 = (3 + 2\sqrt{2}) + (-2\sqrt{2} - 2) - 2 \][/tex]
Primero sumamos y restamos los términos:
[tex]\[ r^2 - 2r - 2 = 3 + 2\sqrt{2} - 2\sqrt{2} - 2 - 2 \][/tex]
Observamos que [tex]\( 2\sqrt{2} \)[/tex] y [tex]\(-2\sqrt{2} \)[/tex] se cancelan entre sí:
[tex]\[ r^2 - 2r - 2 = 3 - 2 - 2 \][/tex]
Finalmente, realizamos la resta:
[tex]\[ r^2 - 2r - 2 = 3 - 4 \][/tex]
[tex]\[ r^2 - 2r - 2 = -1 \][/tex]
Por lo tanto, el valor de la expresión [tex]\( r^2 - 2r - 2 \)[/tex] cuando [tex]\( r = \sqrt{2} + 1 \)[/tex] es [tex]\(-1\)[/tex].
Primero, sabemos que [tex]\( r = \sqrt{2} + 1 \)[/tex]. Queremos encontrar el valor de la expresión [tex]\( r^2 - 2r - 2 \)[/tex].
### Paso 1: Calcula [tex]\( r^2 \)[/tex]
Dado que [tex]\( r = \sqrt{2} + 1 \)[/tex]:
[tex]\[ r^2 = (\sqrt{2} + 1)^2 \][/tex]
Usamos el cuadrado de un binomio:
[tex]\[ r^2 = (\sqrt{2})^2 + 2 \cdot (\sqrt{2}) \cdot 1 + 1^2 \][/tex]
[tex]\[ r^2 = 2 + 2\sqrt{2} + 1 \][/tex]
[tex]\[ r^2 = 3 + 2\sqrt{2} \][/tex]
### Paso 2: Calcula [tex]\(-2r\)[/tex]
Dado que [tex]\( r = \sqrt{2} + 1 \)[/tex]:
[tex]\[ -2r = -2(\sqrt{2} + 1) \][/tex]
[tex]\[ -2r = -2\sqrt{2} - 2 \][/tex]
### Paso 3: Sustituye los valores calculados en la expresión original
Ahora sustituimos [tex]\( r^2 \)[/tex] y [tex]\(-2r\)[/tex] en la expresión [tex]\( r^2 - 2r - 2 \)[/tex]:
[tex]\[ r^2 - 2r - 2 = (3 + 2\sqrt{2}) + (-2\sqrt{2} - 2) - 2 \][/tex]
Primero sumamos y restamos los términos:
[tex]\[ r^2 - 2r - 2 = 3 + 2\sqrt{2} - 2\sqrt{2} - 2 - 2 \][/tex]
Observamos que [tex]\( 2\sqrt{2} \)[/tex] y [tex]\(-2\sqrt{2} \)[/tex] se cancelan entre sí:
[tex]\[ r^2 - 2r - 2 = 3 - 2 - 2 \][/tex]
Finalmente, realizamos la resta:
[tex]\[ r^2 - 2r - 2 = 3 - 4 \][/tex]
[tex]\[ r^2 - 2r - 2 = -1 \][/tex]
Por lo tanto, el valor de la expresión [tex]\( r^2 - 2r - 2 \)[/tex] cuando [tex]\( r = \sqrt{2} + 1 \)[/tex] es [tex]\(-1\)[/tex].
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