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Beliebt Algebra >

faktorisieren 9(p-q)^2+12(p^2-q^2)+4(p+q)^2

Lösung

faktorisieren

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

Lösung

(5 p - q)^{2}

Schritte zur Lösung

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})

Multipliziere aus

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})

Multipliziere aus

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})

Multipliziere aus

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}

Vereinfache

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}

Schreibe

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

um:

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

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

Wende Formel für perfekte quadratische Gleichungen an:

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

a = 5 p, b = q

= (5 p - q)^{2}

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