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

$$( 3 x + y ) ( 3 x - y ) + ( x - y ) ( x + y )$$

Evaluate

$$10x^{2}-2y^{2}$$

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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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}.

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

Combine 9x^{2} and x^{2} to get 10x^{2}.

$$10x^{2}-y^{2}-y^{2}$$

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

$$10x^{2}-2y^{2}$$

Expand

$$10x^{2}-2y^{2}$$

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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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}.

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

Combine 9x^{2} and x^{2} to get 10x^{2}.

$$10x^{2}-y^{2}-y^{2}$$

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

$$10x^{2}-2y^{2}$$