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

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

Evaluate

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

Solution Steps

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

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

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

To find the opposite of x^{2}-y^{2}, find the opposite of each term.

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

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

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

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

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

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

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

Combine 2xy and -2xy to get 0.

$$2y^{2}+x^{2}+y^{2}$$

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

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

Expand

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

Solution Steps

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

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

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

To find the opposite of x^{2}-y^{2}, find the opposite of each term.

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

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

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

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

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

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

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

Combine 2xy and -2xy to get 0.

$$2y^{2}+x^{2}+y^{2}$$

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

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