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

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

Evaluate

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

Solution Steps

Consider \left(3x+y\right)\left(3x-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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

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

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

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

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

Expand

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

Solution Steps

Consider \left(3x+y\right)\left(3x-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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

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

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

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

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