Solve (2a-1)^2-9b^2-(2a-3b+1)^2

$$( 2 a - 1 ) ^ { 2 } - 9 b ^ { 2 } - ( 2 a - 3 b + 1 ) ^ { 2 }$$

Evaluate

$$12ab-8a-18b^{2}+6b$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a-1\right)^{2}.

$$4a^{2}-4a+1-9b^{2}-\left(2a-3b+1\right)^{2}$$

Square 2a-3b+1.

$$4a^{2}-4a+1-9b^{2}-\left(4a^{2}-12ab+4a+9b^{2}-6b+1\right)$$

To find the opposite of 4a^{2}-12ab+4a+9b^{2}-6b+1, find the opposite of each term.

$$4a^{2}-4a+1-9b^{2}-4a^{2}+12ab-4a-9b^{2}+6b-1$$

Combine 4a^{2} and -4a^{2} to get 0.

$$-4a+1-9b^{2}+12ab-4a-9b^{2}+6b-1$$

Combine -4a and -4a to get -8a.

$$-8a+1-9b^{2}+12ab-9b^{2}+6b-1$$

Combine -9b^{2} and -9b^{2} to get -18b^{2}.

$$-8a+1-18b^{2}+12ab+6b-1$$

Subtract 1 from 1 to get 0.

$$-8a-18b^{2}+12ab+6b$$

Expand

$$12ab-8a-18b^{2}+6b$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(2a-1\right)^{2}.

$$4a^{2}-4a+1-9b^{2}-\left(2a-3b+1\right)^{2}$$

Square 2a-3b+1.

$$4a^{2}-4a+1-9b^{2}-\left(4a^{2}-12ab+4a+9b^{2}-6b+1\right)$$

To find the opposite of 4a^{2}-12ab+4a+9b^{2}-6b+1, find the opposite of each term.

$$4a^{2}-4a+1-9b^{2}-4a^{2}+12ab-4a-9b^{2}+6b-1$$

Combine 4a^{2} and -4a^{2} to get 0.

$$-4a+1-9b^{2}+12ab-4a-9b^{2}+6b-1$$

Combine -4a and -4a to get -8a.

$$-8a+1-9b^{2}+12ab-9b^{2}+6b-1$$

Combine -9b^{2} and -9b^{2} to get -18b^{2}.

$$-8a+1-18b^{2}+12ab+6b-1$$

Subtract 1 from 1 to get 0.

$$-8a-18b^{2}+12ab+6b$$