Question

4(x-y)^2-12(x-y)(x+y)+9(x+y)^2 ​

Answer 1

(x + 5 y)^2

Explanation:

Simplify the following:

4 (x - y)^2 - 12 (x - y) (x + y) + 9 (x + y)^2

(x - y) (x - y) = (x) (x) + (x) (-y) + (-y) (x) + (-y) (-y) = x^2 - x y - x y + y^2 = x^2 - 2 x y + y^2:

4 x^2 - 2 x y + y^2 - 12 (x - y) (x + y) + 9 (x + y)^2

4 (x^2 - 2 x y + y^2) = 4 x^2 - 8 x y + 4 y^2:

4 x^2 - 8 x y + 4 y^2 - 12 (x - y) (x + y) + 9 (x + y)^2

(x + y) (x - y) = (x) (x) + (x) (-y) + (y) (x) + (y) (-y) = x^2 - x y + x y - y^2 = x^2 - y^2:

4 x^2 - 8 x y + 4 y^2 - 12 x^2 - y^2 + 9 (x + y)^2

-12 (x^2 - y^2) = 12 y^2 - 12 x^2:

4 x^2 - 8 x y + 4 y^2 + 12 y^2 - 12 x^2 + 9 (x + y)^2

(x + y) (x + y) = (x) (x) + (x) (y) + (y) (x) + (y) (y) = x^2 + x y + x y + y^2 = x^2 + 2 x y + y^2:

4 x^2 - 8 x y + 4 y^2 - 12 x^2 + 12 y^2 + 9 x^2 + 2 x y + y^2

Grouping like terms, 4 x^2 - 8 x y + 4 y^2 - 12 x^2 + 12 y^2 + 9 (x^2 + 2 x y + y^2) = 9 (x^2 + 2 x y + y^2) + (4 y^2 + 12 y^2) - 8 x y + (4 x^2 - 12 x^2):

9 (x^2 + 2 x y + y^2) + (4 y^2 + 12 y^2) - 8 x y + (4 x^2 - 12 x^2)

4 y^2 + 12 y^2 = 16 y^2:

9 (x^2 + 2 x y + y^2) + 16 y^2 - 8 x y + (4 x^2 - 12 x^2)

4 x^2 - 12 x^2 = -8 x^2:

9 (x^2 + 2 x y + y^2) + 16 y^2 - 8 x y + -8 x^2

9 (x^2 + 2 x y + y^2) = 9 x^2 + 18 x y + 9 y^2:

9 x^2 + 18 x y + 9 y^2 + 16 y^2 - 8 x y - 8 x^2

Grouping like terms, 9 x^2 + 18 x y + 9 y^2 + 16 y^2 - 8 x y - 8 x^2 = (9 y^2 + 16 y^2) + (18 x y - 8 x y) + (9 x^2 - 8 x^2):

(9 y^2 + 16 y^2) + (18 x y - 8 x y) + (9 x^2 - 8 x^2)

9 y^2 + 16 y^2 = 25 y^2:

25 y^2 + (18 x y - 8 x y) + (9 x^2 - 8 x^2)

x y 18 + x y (-8) = 10 x y:

25 y^2 + 10 x y + (9 x^2 - 8 x^2)

9 x^2 - 8 x^2 = x^2:

25 y^2 + 10 x y + x^2

The factors of 25 that sum to 10 are 5 and 5. So, 25 y^2 + 10 x y + x^2 = (x + 5 y) (x + 5 y):

(x + 5 y) (x + 5 y)

(x + 5 y) (x + 5 y) = (x + 5 y)^2:

Answer: (x + 5 y)^2

Answer 2

`4(x-y)^2-12(x-y)(x+y)+9(x+y)^2 =\\4(x^2-2xy+y^2)-12(x^2-y^2)+9(x^2+2xy+y^2)=\\4x^2-8xy+4y^2-12x^2+12y^2+9x^2+18xy+9y^2=\\x^2+10xy+25y^2`

which can factorised into
`(x+5y)^2`