Solve (2x+3y)^2-(4x-9y)quad(4x+9y)+(3x-2y)^2

$$( 2 x + 3 y ) ^ { 2 } - ( 4 x - 9 y ) \quad ( 4 x + 9 y ) + ( 3 x - 2 y ) ^ { 2 }$$

Evaluate

$$94y^{2}-3x^{2}$$

Solution Steps

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

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

Consider \left(4x-9y\right)\left(4x+9y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-\left(9y\right)^{2}\right)+\left(3x-2y\right)^{2}$$

Expand \left(9y\right)^{2}.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-9^{2}y^{2}\right)+\left(3x-2y\right)^{2}$$

Calculate 9 to the power of 2 and get 81.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-81y^{2}\right)+\left(3x-2y\right)^{2}$$

To find the opposite of 16x^{2}-81y^{2}, find the opposite of each term.

$$4x^{2}+12xy+9y^{2}-16x^{2}+81y^{2}+\left(3x-2y\right)^{2}$$

Combine 4x^{2} and -16x^{2} to get -12x^{2}.

$$-12x^{2}+12xy+9y^{2}+81y^{2}+\left(3x-2y\right)^{2}$$

Combine 9y^{2} and 81y^{2} to get 90y^{2}.

$$-12x^{2}+12xy+90y^{2}+\left(3x-2y\right)^{2}$$

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

$$-12x^{2}+12xy+90y^{2}+9x^{2}-12xy+4y^{2}$$

Combine -12x^{2} and 9x^{2} to get -3x^{2}.

$$-3x^{2}+12xy+90y^{2}-12xy+4y^{2}$$

Combine 12xy and -12xy to get 0.

$$-3x^{2}+90y^{2}+4y^{2}$$

Combine 90y^{2} and 4y^{2} to get 94y^{2}.

$$-3x^{2}+94y^{2}$$

Expand

$$94y^{2}-3x^{2}$$

Solution Steps

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

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

Consider \left(4x-9y\right)\left(4x+9y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-\left(9y\right)^{2}\right)+\left(3x-2y\right)^{2}$$

Expand \left(9y\right)^{2}.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-9^{2}y^{2}\right)+\left(3x-2y\right)^{2}$$

Calculate 9 to the power of 2 and get 81.

$$4x^{2}+12xy+9y^{2}-\left(16x^{2}-81y^{2}\right)+\left(3x-2y\right)^{2}$$

To find the opposite of 16x^{2}-81y^{2}, find the opposite of each term.

$$4x^{2}+12xy+9y^{2}-16x^{2}+81y^{2}+\left(3x-2y\right)^{2}$$

Combine 4x^{2} and -16x^{2} to get -12x^{2}.

$$-12x^{2}+12xy+9y^{2}+81y^{2}+\left(3x-2y\right)^{2}$$

Combine 9y^{2} and 81y^{2} to get 90y^{2}.

$$-12x^{2}+12xy+90y^{2}+\left(3x-2y\right)^{2}$$

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

$$-12x^{2}+12xy+90y^{2}+9x^{2}-12xy+4y^{2}$$

Combine -12x^{2} and 9x^{2} to get -3x^{2}.

$$-3x^{2}+12xy+90y^{2}-12xy+4y^{2}$$

Combine 12xy and -12xy to get 0.

$$-3x^{2}+90y^{2}+4y^{2}$$

Combine 90y^{2} and 4y^{2} to get 94y^{2}.

$$-3x^{2}+94y^{2}$$