II gimnazjum

Matematyka wokół nas 2

Rozwiązanie 1

Kwiatami nie obsadzono powierzchni, którą można obliczyć jako różnicę pól kwadratu i prostokąta: (2x+4)2−(x+1)⋅(x−2)=(2x+4)⋅(2x+4)−(x2−2x+x−2)=4x2+8x+8x+16−(x2−x−2)=4x2+16x+16−x2+x+2=(3x2+17x+18)m2

Aby obliczyć, ile materiału zostało, wystarczy od pola prostokąta odjąć pole kwadratu. (8a+0,5b)⋅(2b+0,5a)−(2a+b)2=8a⋅2b+8a⋅0,5a+0,5b⋅2b+0,5b⋅0,5a−((2a)2+2⋅2a⋅b+b2)=

Szukane wyrażenia oznaczymy □ oraz ∘ .   a)(4a−□)⋅(5a+2b)=20a2−7ab−∘ 4a⋅5a+4a⋅2b−□⋅5a−□⋅2b=20a2−7ab−∘ 20a2+8ab−5a⋅□−2b⋅□=20a2−7ab−∘ 8ab+□⋅(−5a−2b)=−7ab−∘ □⋅(−5a−2b)=−15ab−∘

Będziemy korzystać ze wzorów skróconego mnożenia (kwadrat sumy i kwadrat różnicy): (a+b)2=a2+2ab+b2 , (a−b)2=a2−2ab+b2 . a)(x+3)2=x2+2⋅x⋅3+32=x2+6x+9

Skorzystamy ze wzoru skróconego mnożenia na kwadrat sumy (a+b)2=a2+2ab+b2 .   a)(3a+2c)2=(3a)2+2⋅3a⋅2c+(2c)2=9a2+12ac+4c2

Skorzystamy ze wzoru skróconego mnożenia na kwadrat różnicy: (a−b)2=a2−2ab+b2 .   a)(3x−8y)2=(3x)2−2⋅3x⋅8y+(8y)2=9x2−48xy+64y2

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Skorzystamy ze wzorów skróconego mnożenia na kwadrat sumy oraz kwadrat różnicy: (a+b)2=a2+2ab+b2,(a−b)2=a2−2ab+b2 .    a)(−3+x)2=(x−3)2=x2−2⋅x⋅3+32=x2−6x+9

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