# How do you factor completely #12x^3y + 6x^2y^2 - 9xy^3#?

##### 2 Answers

#### Explanation:

Given:

#12x^3y+6x^2y^2-9xy^3#

Note that all of the terms are of degree

#12x^3y+6x^2y^2-9xy^3 = 3xy(4x^2+2xy-3y^2)#

We can factor the remaining quadratic by completing the square as follows:

#4x^2+2xy-3y^2 = 1/4(16x^2+8xy-12y^2)#

#color(white)(4x^2+2xy-3y^2) = 1/4((4x)^2+2(4x)(y)+y^2-13y^2)#

#color(white)(4x^2+2xy-3y^2) = 1/4((4x+y)^2-(sqrt(13)y)^2)#

#color(white)(4x^2+2xy-3y^2) = 1/4((4x+y)-sqrt(13)y)((4x+y)+sqrt(13)y)#

#color(white)(4x^2+2xy-3y^2) = 1/4(4x+(1-sqrt(13))y)(4x+(1+sqrt(13))y)#

Putting it all together:

#12x^3y+6x^2y^2-9xy^3 = 3/4xy(4x+(1-sqrt(13))y)(4x+(1+sqrt(13))y)#

#### Explanation:

Making

Solving