Solve (x-y)^2-2(x^2-y^2)+(x+y)^2

$$( x - y ) ^ { 2 } - 2 ( x ^ { 2 } - y ^ { 2 } ) + ( x + y ) ^ { 2 }$$

Evaluate

$$4y^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$-x^{2}-2xy+3y^{2}+x^{2}+2xy+y^{2}$$

Combine -x^{2} and x^{2} to get 0.

$$-2xy+3y^{2}+2xy+y^{2}$$

Combine -2xy and 2xy to get 0.

$$3y^{2}+y^{2}$$

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

$$4y^{2}$$

Expand

$$4y^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

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

$$-x^{2}-2xy+3y^{2}+x^{2}+2xy+y^{2}$$

Combine -x^{2} and x^{2} to get 0.

$$-2xy+3y^{2}+2xy+y^{2}$$

Combine -2xy and 2xy to get 0.

$$3y^{2}+y^{2}$$

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

$$4y^{2}$$