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[tex](\sqrt3x-y)^2-(x-\sqrt3y)^2=\\\\=(\sqrt3x)^2-2\cdot\sqrt3x\cdot y+y^2-(x^2-2\cdot x\cdot\sqrt3y+(\sqrt3y)^2)=\\\\=3x^2-2\sqrt3xy+y^2-(x^2-2\sqrt3xy+3y^2)=\\\\=3x^2-2\sqrt3xy+y^2-x^2+2\sqrt3xy-3y^2=\boxed{2x^2-2y^2}=\\\\=2(\sqrt6+\sqrt2)^2-2(\sqrt6-\sqrt2)^2=\\\\=2((\sqrt6)^2+2\cdot\sqrt6\cdot\sqrt2+(\sqrt2)^2)-2((\sqrt6)^2-2\cdot\sqrt6\cdot\sqrt2+(\sqrt2)^2)=\\\\=2(6+2\sqrt{12}+2)-2(6-2\sqrt{12}+2)=2(8+2\sqrt{4\cdot3})-2(8-2\sqrt{4\cdot3})=\\\\=2(8+4\sqrt3)-2(8-4\sqrt3)=[/tex]
[tex]=16+8\sqrt3-16+8\sqrt3=\boxed{16\sqrt3}\\[/tex]
Wykorzystane wzory skróconego mnożenia
[tex](a-b)^2=a^2-2ab+b^2\\\\(a+b)^2=a^2+2ab+b^2[/tex]
