Solve (2a+3b)^2-(4b-3a)(3a+4b)

$$( 2 a + 3 b ) ^ { 2 } - ( 4 b - 3 a ) ( 3 a + 4 b )$$

Evaluate

$$13a^{2}+12ab-7b^{2}$$

Solution Steps

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

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

Consider \left(4b-3a\right)\left(3a+4b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(4b\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$4a^{2}+12ab+9b^{2}-\left(16b^{2}-\left(3a\right)^{2}\right)$$

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

$$4a^{2}+12ab+9b^{2}-\left(16b^{2}-3^{2}a^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$4a^{2}+12ab+9b^{2}-\left(16b^{2}-9a^{2}\right)$$

To find the opposite of 16b^{2}-9a^{2}, find the opposite of each term.

$$4a^{2}+12ab+9b^{2}-16b^{2}+9a^{2}$$

Combine 9b^{2} and -16b^{2} to get -7b^{2}.

$$4a^{2}+12ab-7b^{2}+9a^{2}$$

Combine 4a^{2} and 9a^{2} to get 13a^{2}.

$$13a^{2}+12ab-7b^{2}$$

Expand

$$13a^{2}+12ab-7b^{2}$$

Solution Steps

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

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

Consider \left(4b-3a\right)\left(3a+4b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(4b\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$4a^{2}+12ab+9b^{2}-\left(16b^{2}-\left(3a\right)^{2}\right)$$

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

$$4a^{2}+12ab+9b^{2}-\left(16b^{2}-3^{2}a^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$4a^{2}+12ab+9b^{2}-\left(16b^{2}-9a^{2}\right)$$

To find the opposite of 16b^{2}-9a^{2}, find the opposite of each term.

$$4a^{2}+12ab+9b^{2}-16b^{2}+9a^{2}$$

Combine 9b^{2} and -16b^{2} to get -7b^{2}.

$$4a^{2}+12ab-7b^{2}+9a^{2}$$

Combine 4a^{2} and 9a^{2} to get 13a^{2}.

$$13a^{2}+12ab-7b^{2}$$