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

Lösung

vereinfachen

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

4 a^{2} + 4 b^{2} + 4 c^{2}

Schritte zur Lösung

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

Schreibe

(a + b + c)^{2}

um:

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

Schreibe

(a - b - c)^{2}

um:

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

Schreibe

(b - c - a)^{2}

um:

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

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

Schreibe

(c - b - a)^{2}

um:

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

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

a^{2} + a^{2} + a^{2} + a^{2} = 4 a^{2}

= 4 a^{2} + 2 ab - 2 ab - 2 ab + 2 ab + 2 a c - 2 a c + 2 a c - 2 a c + b^{2} + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c - 2 b c - 2 b c + c^{2} + c^{2} + c^{2} + c^{2}

Addiere gleiche Elemente:

2 ab - 2 ab - 2 ab + 2 ab = 0

= 4 a^{2} + 2 a c - 2 a c + 2 a c - 2 a c + b^{2} + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c - 2 b c - 2 b c + c^{2} + c^{2} + c^{2} + c^{2}

Addiere gleiche Elemente:

2 a c - 2 a c + 2 a c - 2 a c = 0

= 4 a^{2} + b^{2} + b^{2} + b^{2} + b^{2} + 2 b c + 2 b c - 2 b c - 2 b c + c^{2} + c^{2} + c^{2} + c^{2}

Addiere gleiche Elemente:

b^{2} + b^{2} + b^{2} + b^{2} = 4 b^{2}

= 4 a^{2} + 4 b^{2} + 2 b c + 2 b c - 2 b c - 2 b c + c^{2} + c^{2} + c^{2} + c^{2}

Addiere gleiche Elemente:

2 b c + 2 b c - 2 b c - 2 b c = 0

= 4 a^{2} + 4 b^{2} + c^{2} + c^{2} + c^{2} + c^{2}

Addiere gleiche Elemente:

c^{2} + c^{2} + c^{2} + c^{2} = 4 c^{2}

= 4 a^{2} + 4 b^{2} + 4 c^{2}

s im pl i f y a^{10} a^{8} \cdot b^{3} b b^{2} \cdot c^{2}

s im pl i f y \frac{a ^{3} b ^{4}}{3 a ^{4} b ^{3}}

e x p an d (5 x + 4 y + 3 z)^{2}

f a c t or x^{2} y - y^{2}

s im pl i f y \frac{- 1}{2 u - 4 ( - u - \frac{5}{2} )}