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

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

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 the distributive property to multiply 12 by x-y.

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

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

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

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

$$21x^{2}-18xy+9y^{2}-12y^{2}+4\left(x+y\right)^{2}$$

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

$$21x^{2}-18xy-3y^{2}+4\left(x+y\right)^{2}$$

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

$$21x^{2}-18xy-3y^{2}+4\left(x^{2}+2xy+y^{2}\right)$$

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

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

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

$$25x^{2}-18xy-3y^{2}+8xy+4y^{2}$$

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

$$25x^{2}-10xy-3y^{2}+4y^{2}$$

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

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

Expand

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

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 the distributive property to multiply 12 by x-y.

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

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

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

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

$$21x^{2}-18xy+9y^{2}-12y^{2}+4\left(x+y\right)^{2}$$

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

$$21x^{2}-18xy-3y^{2}+4\left(x+y\right)^{2}$$

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

$$21x^{2}-18xy-3y^{2}+4\left(x^{2}+2xy+y^{2}\right)$$

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

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

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

$$25x^{2}-18xy-3y^{2}+8xy+4y^{2}$$

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

$$25x^{2}-10xy-3y^{2}+4y^{2}$$

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

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