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

$$9 ( x - y ) ^ { 2 } - 12 ( x + y ) ( x - y ) + 4 ( x + y ) ^ { 2 }$$

Evaluate

$$\left(x-5y\right)^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$9\left(x^{2}-2xy+y^{2}\right)-12\left(x+y\right)\left(x-y\right)+4\left(x+y\right)^{2}$$

Use the distributive property to multiply 9 by x^{2}-2xy+y^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(x+y\right)\left(x-y\right)+4\left(x+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(x+y\right)\left(x-y\right)+4\left(x^{2}+2xy+y^{2}\right)$$

Use the distributive property to multiply 4 by x^{2}+2xy+y^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(x+y\right)\left(x-y\right)+4x^{2}+8xy+4y^{2}$$

Use the distributive property to multiply -12 by x+y.

$$9x^{2}-18xy+9y^{2}+\left(-12x-12y\right)\left(x-y\right)+4x^{2}+8xy+4y^{2}$$

Use the distributive property to multiply -12x-12y by x-y and combine like terms.

$$9x^{2}-18xy+9y^{2}-12x^{2}+12y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 9x^{2} and -12x^{2} to get -3x^{2}.

$$-3x^{2}-18xy+9y^{2}+12y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 9y^{2} and 12y^{2} to get 21y^{2}.

$$-3x^{2}-18xy+21y^{2}+4x^{2}+8xy+4y^{2}$$

Combine -3x^{2} and 4x^{2} to get x^{2}.

$$x^{2}-18xy+21y^{2}+8xy+4y^{2}$$

Combine -18xy and 8xy to get -10xy.

$$x^{2}-10xy+21y^{2}+4y^{2}$$

Combine 21y^{2} and 4y^{2} to get 25y^{2}.

$$x^{2}-10xy+25y^{2}$$

Expand

$$x^{2}-10xy+25y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$9\left(x^{2}-2xy+y^{2}\right)-12\left(x+y\right)\left(x-y\right)+4\left(x+y\right)^{2}$$

Use the distributive property to multiply 9 by x^{2}-2xy+y^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(x+y\right)\left(x-y\right)+4\left(x+y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(x+y\right)\left(x-y\right)+4\left(x^{2}+2xy+y^{2}\right)$$

Use the distributive property to multiply 4 by x^{2}+2xy+y^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(x+y\right)\left(x-y\right)+4x^{2}+8xy+4y^{2}$$

Use the distributive property to multiply -12 by x+y.

$$9x^{2}-18xy+9y^{2}+\left(-12x-12y\right)\left(x-y\right)+4x^{2}+8xy+4y^{2}$$

Use the distributive property to multiply -12x-12y by x-y and combine like terms.

$$9x^{2}-18xy+9y^{2}-12x^{2}+12y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 9x^{2} and -12x^{2} to get -3x^{2}.

$$-3x^{2}-18xy+9y^{2}+12y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 9y^{2} and 12y^{2} to get 21y^{2}.

$$-3x^{2}-18xy+21y^{2}+4x^{2}+8xy+4y^{2}$$

Combine -3x^{2} and 4x^{2} to get x^{2}.

$$x^{2}-18xy+21y^{2}+8xy+4y^{2}$$

Combine -18xy and 8xy to get -10xy.

$$x^{2}-10xy+21y^{2}+4y^{2}$$

Combine 21y^{2} and 4y^{2} to get 25y^{2}.

$$x^{2}-10xy+25y^{2}$$