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

$$( a - 3 b ) ( a + 3 b ) + ( 2 b + a ) ( a - 2 b )$$

Evaluate

$$2a^{2}-13b^{2}$$

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

Consider \left(2b+a\right)\left(a-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$2a^{2}-13b^{2}$$

Expand

$$2a^{2}-13b^{2}$$

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

Consider \left(2b+a\right)\left(a-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$2a^{2}-13b^{2}$$