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

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

Evaluate

$$x\left(5x-2y\right)$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}-2xy$$

Expand

$$5x^{2}-2xy$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}-2xy$$