Solve left(x+yright)left(x-yright)+left(x-yright)^2-xleft(x-3yright)

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

Evaluate

$$x\left(x+y\right)$$

Solution Steps

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}.

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

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

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

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

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

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

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

Use the distributive property to multiply x by x-3y.

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

To find the opposite of x^{2}-3xy, find the opposite of each term.

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

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

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

Combine -2xy and 3xy to get xy.

$$x^{2}+xy$$

Expand

$$x^{2}+xy$$

Solution Steps

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}.

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

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

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

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

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

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

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

Use the distributive property to multiply x by x-3y.

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

To find the opposite of x^{2}-3xy, find the opposite of each term.

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

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

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

Combine -2xy and 3xy to get xy.

$$x^{2}+xy$$