Solve (a+12b)(a-12b)-(3a-2b)(-3a-2b)

$$( a + 12 b ) ( a - 12 b ) - ( 3 a - 2 b ) ( - 3 a - 2 b )$$

Evaluate

$$10a^{2}-148b^{2}$$

Solution Steps

Consider \left(a+12b\right)\left(a-12b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$a^{2}-\left(12b\right)^{2}-\left(3a-2b\right)\left(-3a-2b\right)$$

Expand \left(12b\right)^{2}.

$$a^{2}-12^{2}b^{2}-\left(3a-2b\right)\left(-3a-2b\right)$$

Calculate 12 to the power of 2 and get 144.

$$a^{2}-144b^{2}-\left(3a-2b\right)\left(-3a-2b\right)$$

Apply the distributive property by multiplying each term of 3a-2b by each term of -3a-2b.

$$a^{2}-144b^{2}-\left(-9a^{2}-6ab+6ba+4b^{2}\right)$$

Combine -6ab and 6ba to get 0.

$$a^{2}-144b^{2}-\left(-9a^{2}+4b^{2}\right)$$

To find the opposite of -9a^{2}+4b^{2}, find the opposite of each term.

$$a^{2}-144b^{2}-\left(-9a^{2}\right)-4b^{2}$$

The opposite of -9a^{2} is 9a^{2}.

$$a^{2}-144b^{2}+9a^{2}-4b^{2}$$

Combine a^{2} and 9a^{2} to get 10a^{2}.

$$10a^{2}-144b^{2}-4b^{2}$$

Combine -144b^{2} and -4b^{2} to get -148b^{2}.

$$10a^{2}-148b^{2}$$

Expand

$$10a^{2}-148b^{2}$$

Solution Steps

Consider \left(a+12b\right)\left(a-12b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$a^{2}-\left(12b\right)^{2}-\left(3a-2b\right)\left(-3a-2b\right)$$

Expand \left(12b\right)^{2}.

$$a^{2}-12^{2}b^{2}-\left(3a-2b\right)\left(-3a-2b\right)$$

Calculate 12 to the power of 2 and get 144.

$$a^{2}-144b^{2}-\left(3a-2b\right)\left(-3a-2b\right)$$

Apply the distributive property by multiplying each term of 3a-2b by each term of -3a-2b.

$$a^{2}-144b^{2}-\left(-9a^{2}-6ab+6ba+4b^{2}\right)$$

Combine -6ab and 6ba to get 0.

$$a^{2}-144b^{2}-\left(-9a^{2}+4b^{2}\right)$$

To find the opposite of -9a^{2}+4b^{2}, find the opposite of each term.

$$a^{2}-144b^{2}-\left(-9a^{2}\right)-4b^{2}$$

The opposite of -9a^{2} is 9a^{2}.

$$a^{2}-144b^{2}+9a^{2}-4b^{2}$$

Combine a^{2} and 9a^{2} to get 10a^{2}.

$$10a^{2}-144b^{2}-4b^{2}$$

Combine -144b^{2} and -4b^{2} to get -148b^{2}.

$$10a^{2}-148b^{2}$$