展开 (e^{-x^2}-2e^{-x^2}x^2)^2

解答

展开

(e^{- x^{2}} - 2 e^{- x^{2}} x^{2})^{2}

e^{- 2 x^{2}} - 4 e^{- 2 x^{2}} x^{2} + 4 e^{- 2 x^{2}} x^{4}

求解步骤

(e^{- x^{2}} - 2 e^{- x^{2}} x^{2})^{2}

使用完全平方公式:

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

a = e^{- x^{2}}, b = 2 e^{- x^{2}} x^{2}

= (e^{- x^{2}})^{2} - 2 e^{- x^{2}} \cdot 2 e^{- x^{2}} x^{2} + (2 e^{- x^{2}} x^{2})^{2}

化简

(e^{- x^{2}})^{2} - 2 e^{- x^{2}} \cdot 2 e^{- x^{2}} x^{2} + (2 e^{- x^{2}} x^{2})^{2} : e^{- 2 x^{2}} - 4 e^{- 2 x^{2}} x^{2} + 4 e^{- 2 x^{2}} x^{4}

= e^{- 2 x^{2}} - 4 e^{- 2 x^{2}} x^{2} + 4 e^{- 2 x^{2}} x^{4}

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