Solve left(2a+3right)left(2a-3right)+left(9a+bright)left(9a-bright)

$$\left( 2a+3 \right) \left( 2a-3 \right) + \left( 9a+b \right) \left( 9a-b \right)$$

Evaluate

$$85a^{2}-b^{2}-9$$

Solution Steps

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

$$\left(2a\right)^{2}-3^{2}+\left(9a+b\right)\left(9a-b\right)$$

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

$$2^{2}a^{2}-3^{2}+\left(9a+b\right)\left(9a-b\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-3^{2}+\left(9a+b\right)\left(9a-b\right)$$

Calculate 3 to the power of 2 and get 9.

$$4a^{2}-9+\left(9a+b\right)\left(9a-b\right)$$

Consider \left(9a+b\right)\left(9a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4a^{2}-9+\left(9a\right)^{2}-b^{2}$$

Expand \left(9a\right)^{2}.

$$4a^{2}-9+9^{2}a^{2}-b^{2}$$

Calculate 9 to the power of 2 and get 81.

$$4a^{2}-9+81a^{2}-b^{2}$$

Combine 4a^{2} and 81a^{2} to get 85a^{2}.

$$85a^{2}-9-b^{2}$$

Expand

$$85a^{2}-b^{2}-9$$

Solution Steps

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

$$\left(2a\right)^{2}-3^{2}+\left(9a+b\right)\left(9a-b\right)$$

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

$$2^{2}a^{2}-3^{2}+\left(9a+b\right)\left(9a-b\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-3^{2}+\left(9a+b\right)\left(9a-b\right)$$

Calculate 3 to the power of 2 and get 9.

$$4a^{2}-9+\left(9a+b\right)\left(9a-b\right)$$

Consider \left(9a+b\right)\left(9a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4a^{2}-9+\left(9a\right)^{2}-b^{2}$$

Expand \left(9a\right)^{2}.

$$4a^{2}-9+9^{2}a^{2}-b^{2}$$

Calculate 9 to the power of 2 and get 81.

$$4a^{2}-9+81a^{2}-b^{2}$$

Combine 4a^{2} and 81a^{2} to get 85a^{2}.

$$85a^{2}-9-b^{2}$$