Solve (5a-3c)(5a+3c)-(7c-a)(7c+a)

$$( 5 a - 3 c ) ( 5 a + 3 c ) - ( 7 c - a ) ( 7 c + a )$$

Evaluate

$$26a^{2}-58c^{2}$$

Solution Steps

Consider \left(5a-3c\right)\left(5a+3c\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(5a\right)^{2}-\left(3c\right)^{2}-\left(7c-a\right)\left(7c+a\right)$$

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

$$5^{2}a^{2}-\left(3c\right)^{2}-\left(7c-a\right)\left(7c+a\right)$$

Calculate 5 to the power of 2 and get 25.

$$25a^{2}-\left(3c\right)^{2}-\left(7c-a\right)\left(7c+a\right)$$

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

$$25a^{2}-3^{2}c^{2}-\left(7c-a\right)\left(7c+a\right)$$

Calculate 3 to the power of 2 and get 9.

$$25a^{2}-9c^{2}-\left(7c-a\right)\left(7c+a\right)$$

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

$$25a^{2}-9c^{2}-\left(\left(7c\right)^{2}-a^{2}\right)$$

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

$$25a^{2}-9c^{2}-\left(7^{2}c^{2}-a^{2}\right)$$

Calculate 7 to the power of 2 and get 49.

$$25a^{2}-9c^{2}-\left(49c^{2}-a^{2}\right)$$

To find the opposite of 49c^{2}-a^{2}, find the opposite of each term.

$$25a^{2}-9c^{2}-49c^{2}-\left(-a^{2}\right)$$

The opposite of -a^{2} is a^{2}.

$$25a^{2}-9c^{2}-49c^{2}+a^{2}$$

Combine -9c^{2} and -49c^{2} to get -58c^{2}.

$$25a^{2}-58c^{2}+a^{2}$$

Combine 25a^{2} and a^{2} to get 26a^{2}.

$$26a^{2}-58c^{2}$$

Expand

$$26a^{2}-58c^{2}$$

Solution Steps

Consider \left(5a-3c\right)\left(5a+3c\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(5a\right)^{2}-\left(3c\right)^{2}-\left(7c-a\right)\left(7c+a\right)$$

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

$$5^{2}a^{2}-\left(3c\right)^{2}-\left(7c-a\right)\left(7c+a\right)$$

Calculate 5 to the power of 2 and get 25.

$$25a^{2}-\left(3c\right)^{2}-\left(7c-a\right)\left(7c+a\right)$$

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

$$25a^{2}-3^{2}c^{2}-\left(7c-a\right)\left(7c+a\right)$$

Calculate 3 to the power of 2 and get 9.

$$25a^{2}-9c^{2}-\left(7c-a\right)\left(7c+a\right)$$

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

$$25a^{2}-9c^{2}-\left(\left(7c\right)^{2}-a^{2}\right)$$

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

$$25a^{2}-9c^{2}-\left(7^{2}c^{2}-a^{2}\right)$$

Calculate 7 to the power of 2 and get 49.

$$25a^{2}-9c^{2}-\left(49c^{2}-a^{2}\right)$$

To find the opposite of 49c^{2}-a^{2}, find the opposite of each term.

$$25a^{2}-9c^{2}-49c^{2}-\left(-a^{2}\right)$$

The opposite of -a^{2} is a^{2}.

$$25a^{2}-9c^{2}-49c^{2}+a^{2}$$

Combine -9c^{2} and -49c^{2} to get -58c^{2}.

$$25a^{2}-58c^{2}+a^{2}$$

Combine 25a^{2} and a^{2} to get 26a^{2}.

$$26a^{2}-58c^{2}$$