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simplificar (a+b+c)^2+(a-b-c)^2+(-a-b-c)^2

Solución

simplificar

(a + b + c)^{2} + (a - b - c)^{2} + (- a - b - c)^{2}

3 a^{2} + 2 ab + 2 a c + 3 b^{2} + 6 b c + 3 c^{2}

Pasos de solución

(a + b + c)^{2} + (a - b - c)^{2} + (- a - b - c)^{2}

Desarrollar

(a + b + c)^{2} : a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2}

= a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2} + (a - b - c)^{2} + (- a - b - c)^{2}

Desarrollar

(a - b - c)^{2} : a^{2} - 2 ab - 2 a c + b^{2} + 2 b c + c^{2}

= a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2} + a^{2} - 2 ab - 2 a c + b^{2} + 2 b c + c^{2} + (- a - b - c)^{2}

Desarrollar

(- a - b - c)^{2} : a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2}

= a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2} + a^{2} - 2 ab - 2 a c + b^{2} + 2 b c + c^{2} + a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2}

Agrupar términos semejantes

= a^{2} + a^{2} + a^{2} + 2 ab - 2 ab + 2 ab + 2 a c - 2 a c + 2 a c + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c + 2 b c + c^{2} + c^{2} + c^{2}

Sumar elementos similares:

a^{2} + a^{2} + a^{2} = 3 a^{2}

= 3 a^{2} + 2 ab - 2 ab + 2 ab + 2 a c - 2 a c + 2 a c + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c + 2 b c + c^{2} + c^{2} + c^{2}

Sumar elementos similares:

2 ab - 2 ab + 2 ab = 2 ab

= 3 a^{2} + 2 ab + 2 a c - 2 a c + 2 a c + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c + 2 b c + c^{2} + c^{2} + c^{2}

Sumar elementos similares:

2 a c - 2 a c + 2 a c = 2 a c

= 3 a^{2} + 2 ab + 2 a c + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c + 2 b c + c^{2} + c^{2} + c^{2}

Sumar elementos similares:

b^{2} + b^{2} + b^{2} = 3 b^{2}

= 3 a^{2} + 2 ab + 2 a c + 3 b^{2} + 2 b c + 2 b c + 2 b c + c^{2} + c^{2} + c^{2}

Sumar elementos similares:

2 b c + 2 b c + 2 b c = 6 b c

= 3 a^{2} + 2 ab + 2 a c + 3 b^{2} + 6 b c + c^{2} + c^{2} + c^{2}

Sumar elementos similares:

c^{2} + c^{2} + c^{2} = 3 c^{2}

= 3 a^{2} + 2 ab + 2 a c + 3 b^{2} + 6 b c + 3 c^{2}

7 - x^{2} - y^{2} - 2 (x + y)

\frac{1}{2} (x + 3) + \frac{1}{3} (3 x + 6)

(- 3 a^{2} b)^{4}

5.8 \cdot 1 0^{4}

4 (a^{2})