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

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

Evaluate

$$13x^{2}+12xy-7y^{2}$$

Solution Steps

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

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

Consider \left(4y-3x\right)\left(3x+4y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 4 to the power of 2 and get 16.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

To find the opposite of 16y^{2}-9x^{2}, find the opposite of each term.

$$4x^{2}+12xy+9y^{2}-16y^{2}+9x^{2}$$

Combine 9y^{2} and -16y^{2} to get -7y^{2}.

$$4x^{2}+12xy-7y^{2}+9x^{2}$$

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

$$13x^{2}+12xy-7y^{2}$$

Expand

$$13x^{2}+12xy-7y^{2}$$

Solution Steps

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

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

Consider \left(4y-3x\right)\left(3x+4y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 4 to the power of 2 and get 16.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

To find the opposite of 16y^{2}-9x^{2}, find the opposite of each term.

$$4x^{2}+12xy+9y^{2}-16y^{2}+9x^{2}$$

Combine 9y^{2} and -16y^{2} to get -7y^{2}.

$$4x^{2}+12xy-7y^{2}+9x^{2}$$

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

$$13x^{2}+12xy-7y^{2}$$