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

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

Evaluate

$$36b^{2}$$

Solution Steps

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

$$a^{2}+6ab+9b^{2}-2\left(a+3b\right)\left(a-3b\right)+\left(a-3b\right)^{2}$$

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

$$a^{2}+6ab+9b^{2}-2\left(a+3b\right)\left(a-3b\right)+a^{2}-6ab+9b^{2}$$

Use the distributive property to multiply -2 by a+3b.

$$a^{2}+6ab+9b^{2}+\left(-2a-6b\right)\left(a-3b\right)+a^{2}-6ab+9b^{2}$$

Use the distributive property to multiply -2a-6b by a-3b and combine like terms.

$$a^{2}+6ab+9b^{2}-2a^{2}+18b^{2}+a^{2}-6ab+9b^{2}$$

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

$$-a^{2}+6ab+9b^{2}+18b^{2}+a^{2}-6ab+9b^{2}$$

Combine 9b^{2} and 18b^{2} to get 27b^{2}.

$$-a^{2}+6ab+27b^{2}+a^{2}-6ab+9b^{2}$$

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

$$6ab+27b^{2}-6ab+9b^{2}$$

Combine 6ab and -6ab to get 0.

$$27b^{2}+9b^{2}$$

Combine 27b^{2} and 9b^{2} to get 36b^{2}.

$$36b^{2}$$

Expand

$$36b^{2}$$

Solution Steps

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

$$a^{2}+6ab+9b^{2}-2\left(a+3b\right)\left(a-3b\right)+\left(a-3b\right)^{2}$$

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

$$a^{2}+6ab+9b^{2}-2\left(a+3b\right)\left(a-3b\right)+a^{2}-6ab+9b^{2}$$

Use the distributive property to multiply -2 by a+3b.

$$a^{2}+6ab+9b^{2}+\left(-2a-6b\right)\left(a-3b\right)+a^{2}-6ab+9b^{2}$$

Use the distributive property to multiply -2a-6b by a-3b and combine like terms.

$$a^{2}+6ab+9b^{2}-2a^{2}+18b^{2}+a^{2}-6ab+9b^{2}$$

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

$$-a^{2}+6ab+9b^{2}+18b^{2}+a^{2}-6ab+9b^{2}$$

Combine 9b^{2} and 18b^{2} to get 27b^{2}.

$$-a^{2}+6ab+27b^{2}+a^{2}-6ab+9b^{2}$$

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

$$6ab+27b^{2}-6ab+9b^{2}$$

Combine 6ab and -6ab to get 0.

$$27b^{2}+9b^{2}$$

Combine 27b^{2} and 9b^{2} to get 36b^{2}.

$$36b^{2}$$