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受欢迎的代数 >

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

解答

化简

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

解答

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

求解步骤

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

展开

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

展开

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

展开

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

展开

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

对同类项分组

= 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}

同类项相加:

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}

同类项相加:

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}

同类项相加:

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}

同类项相加:

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}

同类项相加:

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}

同类项相加:

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

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

流行的例子

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

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

展开 (5x+4y+3z)^2

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

因式分解 x^2y-y^2

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

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