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

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

Evaluate

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

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

Combine -4b^{2} and -4b^{2} to get -8b^{2}.

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

Expand

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

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

Combine -4b^{2} and -4b^{2} to get -8b^{2}.

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