Задание 469

Решение:

• \({\largeа)}\ (a+b)^2-c^2=(a+b+c)(a+b-c)\)

• \({\largeб)}\ (a-b)^2-c^2=(a-b+c)(a-b-c)\)

• \({\largeв)}\ (x-y)^2-(x+y)^2=((x-y)+(x+y))((x-y)-(x+y))=(x-y+x+y)(x-y-x-y)=2x\cdot({-}2y)={-}4xy\)

• \({\largeг)}\ (a+b)^2-(x+y)^2=((a+b)+(x+y))((a+b)-(x+y))=(a+b+x+y)(a+b-x-y)\)

• \({\largeд)}\ (2x-y)^2-(3x-2y)^2=((2x-y)+(3x-2y))((2x-y)-(3x-2y))=(2x-y+3x-2y)(2x-y-3x+2y)=(5x-3y)(y-x)\)

• \({\largeе)}\ (m^2-4n)^2-(m^2-2n)^2=((m^2-4n)+(m^2-2n))((m^2-4n)-(m^2-2n))=(m^2-4n+m^2-2n)(m^2-4n-m^2+2n)=(2m^2-6n)({-}2n)=2(m^2-3n)({-}2n)={-}4n(m^2-3n)\)

• \({\largeж)}\ (a+b)^2+2(a+b)+1=(a+b)^2+2\cdot(a+b)\cdot1+1^2=(a+b+1)^2=(a+b+1)(a+b+1)\)

• \({\largeз)}\ (x-2y)^2+4(x-2y)+4=(x-2y)^2+2\cdot(x-2y)\cdot2+2^2=(x-2y+2)^2=(x-2y+2)(x-2y+2)\)

• \({\largeи)}\ 9a^2-6a(a+1)+(a+1)^2=(3a)^2-2\cdot3a\cdot(a+1)+(a+1)^2=(3a-(a+1))^2=(3a-a-1)^2=(2a-1)^2=(2a-1)(2a-1)\)

• \({\largeк)}\ 16m^2-8m(3-m)+(3-m)^2=(4m)^2-2\cdot4m\cdot(3-m)+(3-m)^2=(4m-(3-m))^2=(4m-3+m)^2=(5m-3)^2=(5m-3)(5m-3)\)