Solve (5x-7y)(5x+7y)+(7x-5y)(7x+5y)

$$( 5 x - 7 y ) ( 5 x + 7 y ) + ( 7 x - 5 y ) ( 7 x + 5 y )$$

Evaluate

$$74\left(x^{2}-y^{2}\right)$$

Solution Steps

Consider \left(5x-7y\right)\left(5x+7y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(5x\right)^{2}-\left(7y\right)^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

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

$$5^{2}x^{2}-\left(7y\right)^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-\left(7y\right)^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Expand \left(7y\right)^{2}.

$$25x^{2}-7^{2}y^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Calculate 7 to the power of 2 and get 49.

$$25x^{2}-49y^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Consider \left(7x-5y\right)\left(7x+5y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$25x^{2}-49y^{2}+\left(7x\right)^{2}-\left(5y\right)^{2}$$

Expand \left(7x\right)^{2}.

$$25x^{2}-49y^{2}+7^{2}x^{2}-\left(5y\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$25x^{2}-49y^{2}+49x^{2}-\left(5y\right)^{2}$$

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

$$25x^{2}-49y^{2}+49x^{2}-5^{2}y^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-49y^{2}+49x^{2}-25y^{2}$$

Combine 25x^{2} and 49x^{2} to get 74x^{2}.

$$74x^{2}-49y^{2}-25y^{2}$$

Combine -49y^{2} and -25y^{2} to get -74y^{2}.

$$74x^{2}-74y^{2}$$

Expand

$$74x^{2}-74y^{2}$$

Solution Steps

Consider \left(5x-7y\right)\left(5x+7y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(5x\right)^{2}-\left(7y\right)^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

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

$$5^{2}x^{2}-\left(7y\right)^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-\left(7y\right)^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Expand \left(7y\right)^{2}.

$$25x^{2}-7^{2}y^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Calculate 7 to the power of 2 and get 49.

$$25x^{2}-49y^{2}+\left(7x-5y\right)\left(7x+5y\right)$$

Consider \left(7x-5y\right)\left(7x+5y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$25x^{2}-49y^{2}+\left(7x\right)^{2}-\left(5y\right)^{2}$$

Expand \left(7x\right)^{2}.

$$25x^{2}-49y^{2}+7^{2}x^{2}-\left(5y\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$25x^{2}-49y^{2}+49x^{2}-\left(5y\right)^{2}$$

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

$$25x^{2}-49y^{2}+49x^{2}-5^{2}y^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}-49y^{2}+49x^{2}-25y^{2}$$

Combine 25x^{2} and 49x^{2} to get 74x^{2}.

$$74x^{2}-49y^{2}-25y^{2}$$

Combine -49y^{2} and -25y^{2} to get -74y^{2}.

$$74x^{2}-74y^{2}$$