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EJERCICIO: 01
Si [tex]$a + b = 0$[/tex] y [tex]$ab = 3$[/tex], determinar:

[tex]$a^2 + b^2 + 3a + 3b$[/tex]

Sagot :

GinnyAnswer
Para determinar [tex]\( a^2 + b^2 + 3a + 3b \)[/tex] dados los valores [tex]\( a + b = 0 \)[/tex] y [tex]\( ab = 3 \)[/tex], sigamos los siguientes pasos detalladamente:

1. Comencemos con las ecuaciones dadas:
- [tex]\( a + b = 0 \)[/tex]
- [tex]\( ab = 3 \)[/tex]

2. Calculemos [tex]\( a^2 + b^2 \)[/tex] utilizando una identidad algebraica:
- Sabemos que:
[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]
- Dado que [tex]\( a + b = 0 \)[/tex], podemos sustituir en la identidad:
[tex]\[ (0)^2 = a^2 + b^2 + 2ab \][/tex]
- Esto se simplifica a:
[tex]\[ 0 = a^2 + b^2 + 2ab \][/tex]

3. Sustituir el valor de [tex]\( ab \)[/tex] en la ecuación anterior:
- Dado que [tex]\( ab = 3 \)[/tex]:
[tex]\[ 0 = a^2 + b^2 + 2(3) \][/tex]
- Esto se simplifica a:
[tex]\[ 0 = a^2 + b^2 + 6 \][/tex]
- Resolviendo para [tex]\( a^2 + b^2 \)[/tex], obtenemos:
[tex]\[ a^2 + b^2 = -6 \][/tex]

4. Calculemos el término [tex]\( 3a + 3b \)[/tex]:
- Dado que [tex]\( a + b = 0 \)[/tex], multiplicamos ambos lados por 3:
[tex]\[ 3a + 3b = 3 \times 0 = 0 \][/tex]

5. Sumemos los resultados obtenidos:
- Tenemos que [tex]\( a^2 + b^2 = -6 \)[/tex] y [tex]\( 3a + 3b = 0 \)[/tex].
- Entonces:
[tex]\[ a^2 + b^2 + 3a + 3b = -6 + 0 = -6 \][/tex]

Por tanto, el valor de [tex]\( a^2 + b^2 + 3a + 3b \)[/tex] es [tex]\(\boxed{-6}\)[/tex].
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