因式分解 a^2+2a(a+b)+(a+b)^2

解答

因式分解

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

(2 a + b)^{2}

求解步骤

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

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

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

乘开

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

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

化简

a^{2} + 2 a^{2} + 2 ab + a^{2} + 2 ab + b^{2} : 4 a^{2} + 4 ab + b^{2}

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

将

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

改写为

(2 a)^{2} + 2 \cdot 2 ab + b^{2}

= (2 a)^{2} + 2 \cdot 2 ab + b^{2}

使用完全平方公式:

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

a = 2 a, b = b

= (2 a + b)^{2}

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