Solve (x-2y)^2+(x-3y)(x+3y)-(x+2y)(x-3y)

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

Evaluate

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

Solution Steps

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

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

Consider \left(x-3y\right)\left(x+3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3y\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

$$x^{2}-4xy+4y^{2}+x^{2}-9y^{2}-\left(x+2y\right)\left(x-3y\right)$$

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

$$2x^{2}-4xy+4y^{2}-9y^{2}-\left(x+2y\right)\left(x-3y\right)$$

Combine 4y^{2} and -9y^{2} to get -5y^{2}.

$$2x^{2}-4xy-5y^{2}-\left(x+2y\right)\left(x-3y\right)$$

Use the distributive property to multiply x+2y by x-3y and combine like terms.

$$2x^{2}-4xy-5y^{2}-\left(x^{2}-xy-6y^{2}\right)$$

To find the opposite of x^{2}-xy-6y^{2}, find the opposite of each term.

$$2x^{2}-4xy-5y^{2}-x^{2}+xy+6y^{2}$$

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

$$x^{2}-4xy-5y^{2}+xy+6y^{2}$$

Combine -4xy and xy to get -3xy.

$$x^{2}-3xy-5y^{2}+6y^{2}$$

Combine -5y^{2} and 6y^{2} to get y^{2}.

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

Expand

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

Solution Steps

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

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

Consider \left(x-3y\right)\left(x+3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3y\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

$$x^{2}-4xy+4y^{2}+x^{2}-9y^{2}-\left(x+2y\right)\left(x-3y\right)$$

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

$$2x^{2}-4xy+4y^{2}-9y^{2}-\left(x+2y\right)\left(x-3y\right)$$

Combine 4y^{2} and -9y^{2} to get -5y^{2}.

$$2x^{2}-4xy-5y^{2}-\left(x+2y\right)\left(x-3y\right)$$

Use the distributive property to multiply x+2y by x-3y and combine like terms.

$$2x^{2}-4xy-5y^{2}-\left(x^{2}-xy-6y^{2}\right)$$

To find the opposite of x^{2}-xy-6y^{2}, find the opposite of each term.

$$2x^{2}-4xy-5y^{2}-x^{2}+xy+6y^{2}$$

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

$$x^{2}-4xy-5y^{2}+xy+6y^{2}$$

Combine -4xy and xy to get -3xy.

$$x^{2}-3xy-5y^{2}+6y^{2}$$

Combine -5y^{2} and 6y^{2} to get y^{2}.

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