已知x=√3+√2,y=√3-√2,求代数式3x²-5xy+3y²的值

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已知x=√3+√2,y=√3-√2,求代数式3x²-5xy+3y²的值
已知x=√3+√2,y=√3-√2,求代数式3x²-5xy+3y²的值

已知x=√3+√2,y=√3-√2,求代数式3x²-5xy+3y²的值
3x²-5xy+3y²
=3x²-6xy+3y²+xy
=3(x²-2xy+y²)+xy
=3(x-y)²+xy
=3[√3+√2-(√3-√2)]^2+(√3+√2)(√3-√2)
=3[√3+√2-√3+√2]^2+1
=3(2√2)^2+1
=3*8+1
=24+1
=25

3x^2-5xy+3y^2
=3(x+y)^2-6xy-5xy
=3(x+y)^2-11xy
x+y=2根号3 xy=3-2=1
代入得：
原式=3*(2根号3)^2-11=3*4*3-11=36-11=25

3x²-5xy+3y²=3(x-y)²+xy=3(2√2)²+3-2=23

见图

3x²-5xy+3y²
=3 x²+3y²+6xy-11 xy
=3(x²+y²+2xy)-11xy
=3(x+y)^2-11xy
=3(√3+√2+√3-√2)^2-11(√3+√2)(√3-√2,)
=3(2√3)^2-11(3-2)^2
=36-11
=25

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