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

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

Evaluate

$$\left(a-4b\right)\left(3a-2b\right)$$

Solution Steps

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

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

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

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

To find the opposite of b^{2}+2ba+a^{2}, find the opposite of each term.

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

Combine 9b^{2} and -b^{2} to get 8b^{2}.

$$4a^{2}-12ab+8b^{2}-2ba-a^{2}$$

Combine -12ab and -2ba to get -14ab.

$$4a^{2}-14ab+8b^{2}-a^{2}$$

Combine 4a^{2} and -a^{2} to get 3a^{2}.

$$3a^{2}-14ab+8b^{2}$$

Expand

$$3a^{2}-14ab+8b^{2}$$

Solution Steps

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

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

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

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

To find the opposite of b^{2}+2ba+a^{2}, find the opposite of each term.

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

Combine 9b^{2} and -b^{2} to get 8b^{2}.

$$4a^{2}-12ab+8b^{2}-2ba-a^{2}$$

Combine -12ab and -2ba to get -14ab.

$$4a^{2}-14ab+8b^{2}-a^{2}$$

Combine 4a^{2} and -a^{2} to get 3a^{2}.

$$3a^{2}-14ab+8b^{2}$$