Solve (2a^2-3)^2-(2a^2+3b)(2a^2-3b)

$$( 2 a ^ { 2 } - 3 ) ^ { 2 } - ( 2 a ^ { 2 } + 3 b ) ( 2 a ^ { 2 } - 3 b )$$

Evaluate

$$9+9b^{2}-12a^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a^{2}-3\right)^{2}.

$$4\left(a^{2}\right)^{2}-12a^{2}+9-\left(2a^{2}+3b\right)\left(2a^{2}-3b\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4a^{4}-12a^{2}+9-\left(2a^{2}+3b\right)\left(2a^{2}-3b\right)$$

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

$$4a^{4}-12a^{2}+9-\left(\left(2a^{2}\right)^{2}-\left(3b\right)^{2}\right)$$

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

$$4a^{4}-12a^{2}+9-\left(2^{2}\left(a^{2}\right)^{2}-\left(3b\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4a^{4}-12a^{2}+9-\left(2^{2}a^{4}-\left(3b\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{4}-12a^{2}+9-\left(4a^{4}-\left(3b\right)^{2}\right)$$

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

$$4a^{4}-12a^{2}+9-\left(4a^{4}-3^{2}b^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$4a^{4}-12a^{2}+9-\left(4a^{4}-9b^{2}\right)$$

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

$$4a^{4}-12a^{2}+9-4a^{4}+9b^{2}$$

Combine 4a^{4} and -4a^{4} to get 0.

$$-12a^{2}+9+9b^{2}$$

Expand

$$9+9b^{2}-12a^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a^{2}-3\right)^{2}.

$$4\left(a^{2}\right)^{2}-12a^{2}+9-\left(2a^{2}+3b\right)\left(2a^{2}-3b\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4a^{4}-12a^{2}+9-\left(2a^{2}+3b\right)\left(2a^{2}-3b\right)$$

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

$$4a^{4}-12a^{2}+9-\left(\left(2a^{2}\right)^{2}-\left(3b\right)^{2}\right)$$

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

$$4a^{4}-12a^{2}+9-\left(2^{2}\left(a^{2}\right)^{2}-\left(3b\right)^{2}\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$4a^{4}-12a^{2}+9-\left(2^{2}a^{4}-\left(3b\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{4}-12a^{2}+9-\left(4a^{4}-\left(3b\right)^{2}\right)$$

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

$$4a^{4}-12a^{2}+9-\left(4a^{4}-3^{2}b^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$4a^{4}-12a^{2}+9-\left(4a^{4}-9b^{2}\right)$$

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

$$4a^{4}-12a^{2}+9-4a^{4}+9b^{2}$$

Combine 4a^{4} and -4a^{4} to get 0.

$$-12a^{2}+9+9b^{2}$$