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

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

Evaluate

$$2y\left(3y-2x\right)$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

To find the opposite of x^{2}-4y^{2}, find the opposite of each term.

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

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

$$-4xy+4y^{2}+4y^{2}-2y^{2}$$

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

$$-4xy+8y^{2}-2y^{2}$$

Combine 8y^{2} and -2y^{2} to get 6y^{2}.

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

Expand

$$6y^{2}-4xy$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

To find the opposite of x^{2}-4y^{2}, find the opposite of each term.

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

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

$$-4xy+4y^{2}+4y^{2}-2y^{2}$$

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

$$-4xy+8y^{2}-2y^{2}$$

Combine 8y^{2} and -2y^{2} to get 6y^{2}.

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