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

$$( x - 3 y ) ( 3 y + x ) - ( 2 y - 3 x ) ( 3 x + 2 y )$$

Evaluate

$$10x^{2}-13y^{2}$$

Solution Steps

Consider \left(x-3y\right)\left(3y+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 3 to the power of 2 and get 9.

$$x^{2}-9y^{2}-\left(2y-3x\right)\left(3x+2y\right)$$

Consider \left(2y-3x\right)\left(3x+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}.

$$x^{2}-9y^{2}-\left(\left(2y\right)^{2}-\left(3x\right)^{2}\right)$$

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

$$x^{2}-9y^{2}-\left(2^{2}y^{2}-\left(3x\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-9y^{2}-\left(4y^{2}-\left(3x\right)^{2}\right)$$

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

$$x^{2}-9y^{2}-\left(4y^{2}-3^{2}x^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

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

To find the opposite of 4y^{2}-9x^{2}, find the opposite of each term.

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

The opposite of -9x^{2} is 9x^{2}.

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

Combine -9y^{2} and -4y^{2} to get -13y^{2}.

$$x^{2}-13y^{2}+9x^{2}$$

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

$$10x^{2}-13y^{2}$$

Expand

$$10x^{2}-13y^{2}$$

Solution Steps

Consider \left(x-3y\right)\left(3y+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 3 to the power of 2 and get 9.

$$x^{2}-9y^{2}-\left(2y-3x\right)\left(3x+2y\right)$$

Consider \left(2y-3x\right)\left(3x+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}.

$$x^{2}-9y^{2}-\left(\left(2y\right)^{2}-\left(3x\right)^{2}\right)$$

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

$$x^{2}-9y^{2}-\left(2^{2}y^{2}-\left(3x\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-9y^{2}-\left(4y^{2}-\left(3x\right)^{2}\right)$$

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

$$x^{2}-9y^{2}-\left(4y^{2}-3^{2}x^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

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

To find the opposite of 4y^{2}-9x^{2}, find the opposite of each term.

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

The opposite of -9x^{2} is 9x^{2}.

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

Combine -9y^{2} and -4y^{2} to get -13y^{2}.

$$x^{2}-13y^{2}+9x^{2}$$

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

$$10x^{2}-13y^{2}$$