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

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

Evaluate

$$3\left(x-3\right)\left(x-1\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.

$$4x^{2}-12x+9-\left(x+y\right)\left(x-y\right)-y^{2}$$

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}.

$$4x^{2}-12x+9-\left(x^{2}-y^{2}\right)-y^{2}$$

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

$$4x^{2}-12x+9-x^{2}+y^{2}-y^{2}$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-12x+9+y^{2}-y^{2}$$

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

$$3x^{2}-12x+9$$

Expand

$$3x^{2}-12x+9$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.

$$4x^{2}-12x+9-\left(x+y\right)\left(x-y\right)-y^{2}$$

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}.

$$4x^{2}-12x+9-\left(x^{2}-y^{2}\right)-y^{2}$$

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

$$4x^{2}-12x+9-x^{2}+y^{2}-y^{2}$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}-12x+9+y^{2}-y^{2}$$

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

$$3x^{2}-12x+9$$