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

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

Evaluate

$$13y^{2}-12xy-5x^{2}$$

Solution Steps

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

$$4x^{2}-12xy+9y^{2}-\left(2y+3x\right)\left(3x-2y\right)$$

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}.

$$4x^{2}-12xy+9y^{2}-\left(\left(3x\right)^{2}-\left(2y\right)^{2}\right)$$

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

$$4x^{2}-12xy+9y^{2}-\left(3^{2}x^{2}-\left(2y\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$-5x^{2}-12xy+9y^{2}+4y^{2}$$

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

$$-5x^{2}-12xy+13y^{2}$$

Expand

$$13y^{2}-12xy-5x^{2}$$

Solution Steps

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

$$4x^{2}-12xy+9y^{2}-\left(2y+3x\right)\left(3x-2y\right)$$

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}.

$$4x^{2}-12xy+9y^{2}-\left(\left(3x\right)^{2}-\left(2y\right)^{2}\right)$$

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

$$4x^{2}-12xy+9y^{2}-\left(3^{2}x^{2}-\left(2y\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$-5x^{2}-12xy+9y^{2}+4y^{2}$$

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

$$-5x^{2}-12xy+13y^{2}$$