因数 9(p-q)^2+12(p^2-q^2)+4(p+q)^2

解

因数

9 (p - q)^{2} + 12 (p^{2} - q^{2}) + 4 (p + q)^{2}

(5 p - q)^{2}

解答ステップ

9 (p - q)^{2} + 12 (p^{2} - q^{2}) + 4 (p + q)^{2}

12 (p^{2} - q^{2}) = 12 (p + q) (p - q)

= 9 (p - q)^{2} + 12 (p + q) (p - q) + 4 (p + q)^{2}

(p - q)^{2} : p^{2} - 2 pq + q^{2}

= 9 (p^{2} - 2 pq + q^{2}) + 12 (p + q) (p - q) + 4 (p + q)^{2}

(p + q)^{2} : p^{2} + 2 pq + q^{2}

= 9 (p^{2} - 2 pq + q^{2}) + 12 (p + q) (p - q) + 4 (p^{2} + 2 pq + q^{2})

拡張

9 (p^{2} - 2 pq + q^{2}) : 9 p^{2} - 18 pq + 9 q^{2}

= 9 p^{2} - 18 pq + 9 q^{2} + 12 (p + q) (p - q) + 4 (p^{2} + 2 pq + q^{2})

拡張

12 (p + q) (p - q) : 12 p^{2} - 12 q^{2}

= 9 p^{2} - 18 pq + 9 q^{2} + 12 p^{2} - 12 q^{2} + 4 (p^{2} + 2 pq + q^{2})

拡張

4 (p^{2} + 2 pq + q^{2}) : 4 p^{2} + 8 pq + 4 q^{2}

= 9 p^{2} - 18 pq + 9 q^{2} + 12 p^{2} - 12 q^{2} + 4 p^{2} + 8 pq + 4 q^{2}

簡素化

9 p^{2} - 18 pq + 9 q^{2} + 12 p^{2} - 12 q^{2} + 4 p^{2} + 8 pq + 4 q^{2} : 25 p^{2} - 10 pq + q^{2}

= 25 p^{2} - 10 pq + q^{2}

25 p^{2} - 10 pq + q^{2}

を書き換え

(5 p)^{2} - 2 \cdot 5 pq + q^{2}

= (5 p)^{2} - 2 \cdot 5 pq + q^{2}

完全平方式を適用する:

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

a = 5 p, b = q

= (5 p - q)^{2}