Solve (2x-3y)^2-(x+2y)(x-2y)+4y(3x-y)

$$( 2 x - 3 y ) ^ { 2 } - ( x + 2 y ) ( x - 2 y ) + 4 y ( 3 x - y )$$

Evaluate

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

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

$$3x^{2}-12xy+13y^{2}+4y\left(3x-y\right)$$

Use the distributive property to multiply 4y by 3x-y.

$$3x^{2}-12xy+13y^{2}+12yx-4y^{2}$$

Combine -12xy and 12yx to get 0.

$$3x^{2}+13y^{2}-4y^{2}$$

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

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

Expand

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

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

$$3x^{2}-12xy+13y^{2}+4y\left(3x-y\right)$$

Use the distributive property to multiply 4y by 3x-y.

$$3x^{2}-12xy+13y^{2}+12yx-4y^{2}$$

Combine -12xy and 12yx to get 0.

$$3x^{2}+13y^{2}-4y^{2}$$

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

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