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Sagot :
Para encontrar el valor de \(a^2 + b^2\) dado que \(a = 3x - 5\) y \(b = 5x + 2\), sigamos los siguientes pasos detallados:
1. Expresar \(a\) y \(b\):
[tex]\[ a = 3x - 5 \][/tex]
[tex]\[ b = 5x + 2 \][/tex]
2. Calcular \(a^2\):
[tex]\[ a^2 = (3x - 5)^2 \][/tex]
[tex]\[ a^2 = (3x - 5)(3x - 5) = 9x^2 - 15x - 15x + 25 = 9x^2 - 30x + 25 \][/tex]
3. Calcular \(b^2\):
[tex]\[ b^2 = (5x + 2)^2 \][/tex]
[tex]\[ b^2 = (5x + 2)(5x + 2) = 25x^2 + 10x + 10x + 4 = 25x^2 + 20x + 4 \][/tex]
4. Sumar \(a^2\) y \(b^2\):
[tex]\[ a^2 + b^2 = (9x^2 - 30x + 25) + (25x^2 + 20x + 4) \][/tex]
[tex]\[ a^2 + b^2 = 9x^2 + 25x^2 - 30x + 20x + 25 + 4 \][/tex]
[tex]\[ a^2 + b^2 = 34x^2 - 10x + 29 \][/tex]
Por lo tanto, el valor de \(a^2 + b^2\) es:
[tex]\[ 34x^2 - 10x + 29 \][/tex]
La respuesta correcta es:
[tex]\[ \boxed{34x^2 - 10x + 29} \][/tex]
1. Expresar \(a\) y \(b\):
[tex]\[ a = 3x - 5 \][/tex]
[tex]\[ b = 5x + 2 \][/tex]
2. Calcular \(a^2\):
[tex]\[ a^2 = (3x - 5)^2 \][/tex]
[tex]\[ a^2 = (3x - 5)(3x - 5) = 9x^2 - 15x - 15x + 25 = 9x^2 - 30x + 25 \][/tex]
3. Calcular \(b^2\):
[tex]\[ b^2 = (5x + 2)^2 \][/tex]
[tex]\[ b^2 = (5x + 2)(5x + 2) = 25x^2 + 10x + 10x + 4 = 25x^2 + 20x + 4 \][/tex]
4. Sumar \(a^2\) y \(b^2\):
[tex]\[ a^2 + b^2 = (9x^2 - 30x + 25) + (25x^2 + 20x + 4) \][/tex]
[tex]\[ a^2 + b^2 = 9x^2 + 25x^2 - 30x + 20x + 25 + 4 \][/tex]
[tex]\[ a^2 + b^2 = 34x^2 - 10x + 29 \][/tex]
Por lo tanto, el valor de \(a^2 + b^2\) es:
[tex]\[ 34x^2 - 10x + 29 \][/tex]
La respuesta correcta es:
[tex]\[ \boxed{34x^2 - 10x + 29} \][/tex]
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