Question

在实数范围内分解因式：3x²+4xy-2y²

Answer 1

3x²+4xy-2y²
=(1/3)*(9x²+12xy-6y²)
=(1/3)*((3x)²+2*(3x)*(2y)+(2y)²-(2y)²-6y²)
=(1/3)*((3x+2y)²-10y²)
=(1/3)(3x+2y+√10y)(3x+2y-√10y)

=====================================

3x²+4xy-2y²
=(√3x)²+2*(√3x)(2/√3y)+(2/√3y)²-(10/3)*y²
=(√3x+(2/√3)y)²-(10/3)*y²
=(√3x+(2/√3)y+(√10/√3)y)(√3x+(2/√3)y-(√10/√3)y)
=(1/√3*1/√3)*(3x+2y+√10y)(3x+2y-√10y)
=(1/3)(3x+2y+√10y)(3x+2y-√10y)

Answer 2

配方法
3x²+4xy-2y²=3[x²+(4/3)xy+(2y/3)²-(2y/3)²-(2/3)y²]\

=3{ [x+(2/3)y]²-[(√10/3)y]² }
=3[x+(2/3)y+(√10/3)y][x+(2/3)y-(√10/3)y]
=[3x+(2+√10)y][3x+(2-√10)y]/3

Answer 3

3x²+4xy-2y²
=3x²-2(-2xy+y²)
=3x²-2(x²-2xy+y²)+2x²
=5x²-2(x-y)²
=(根号5*x+根号2*x-根号2*y)(根号5*x-根号2*x+根号2*y)