Solve left(2x-3yright)left(3y+2xright)-left(4y-3xright)left(3x+4yright)

$$\left( 2x-3y \right) \left( 3y+2x \right) - \left( 4y-3x \right) \left( 3x+4y \right)$$

Evaluate

$$13x^{2}-25y^{2}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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}.

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

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

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

Calculate 4 to the power of 2 and get 16.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

Combine -9y^{2} and -16y^{2} to get -25y^{2}.

$$4x^{2}-25y^{2}+9x^{2}$$

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

$$13x^{2}-25y^{2}$$

Expand

$$13x^{2}-25y^{2}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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}.

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

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

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

Calculate 4 to the power of 2 and get 16.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

Combine -9y^{2} and -16y^{2} to get -25y^{2}.

$$4x^{2}-25y^{2}+9x^{2}$$

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

$$13x^{2}-25y^{2}$$