Solve (5x-y)(y+5x)-(3y+2x)(3y-2x)

$$( 5 x - y ) ( y + 5 x ) - ( 3 y + 2 x ) ( 3 y - 2 x )$$

Evaluate

$$29x^{2}-10y^{2}$$

Solution Steps

Consider \left(5x-y\right)\left(y+5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(5x\right)^{2}-y^{2}-\left(3y+2x\right)\left(3y-2x\right)$$

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-y^{2}-\left(3y+2x\right)\left(3y-2x\right)$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-y^{2}-\left(3y+2x\right)\left(3y-2x\right)$$

Consider \left(3y+2x\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}.

$$25x^{2}-y^{2}-\left(\left(3y\right)^{2}-\left(2x\right)^{2}\right)$$

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

$$25x^{2}-y^{2}-\left(3^{2}y^{2}-\left(2x\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}-y^{2}-\left(9y^{2}-\left(2x\right)^{2}\right)$$

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

$$25x^{2}-y^{2}-\left(9y^{2}-2^{2}x^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-y^{2}-\left(9y^{2}-4x^{2}\right)$$

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

$$25x^{2}-y^{2}-9y^{2}-\left(-4x^{2}\right)$$

The opposite of -4x^{2} is 4x^{2}.

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

Combine -y^{2} and -9y^{2} to get -10y^{2}.

$$25x^{2}-10y^{2}+4x^{2}$$

Combine 25x^{2} and 4x^{2} to get 29x^{2}.

$$29x^{2}-10y^{2}$$

Expand

$$29x^{2}-10y^{2}$$

Solution Steps

Consider \left(5x-y\right)\left(y+5x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(5x\right)^{2}-y^{2}-\left(3y+2x\right)\left(3y-2x\right)$$

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}-y^{2}-\left(3y+2x\right)\left(3y-2x\right)$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-y^{2}-\left(3y+2x\right)\left(3y-2x\right)$$

Consider \left(3y+2x\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}.

$$25x^{2}-y^{2}-\left(\left(3y\right)^{2}-\left(2x\right)^{2}\right)$$

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

$$25x^{2}-y^{2}-\left(3^{2}y^{2}-\left(2x\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$25x^{2}-y^{2}-\left(9y^{2}-\left(2x\right)^{2}\right)$$

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

$$25x^{2}-y^{2}-\left(9y^{2}-2^{2}x^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-y^{2}-\left(9y^{2}-4x^{2}\right)$$

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

$$25x^{2}-y^{2}-9y^{2}-\left(-4x^{2}\right)$$

The opposite of -4x^{2} is 4x^{2}.

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

Combine -y^{2} and -9y^{2} to get -10y^{2}.

$$25x^{2}-10y^{2}+4x^{2}$$

Combine 25x^{2} and 4x^{2} to get 29x^{2}.

$$29x^{2}-10y^{2}$$