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

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

Evaluate

$$-8a^{2}$$

Solution Steps

Consider \left(a+2b\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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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}.

$$a^{2}-4b^{2}-\left(\left(3a\right)^{2}-\left(2b\right)^{2}\right)$$

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

$$a^{2}-4b^{2}-\left(3^{2}a^{2}-\left(2b\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

The opposite of -4b^{2} is 4b^{2}.

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

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

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

Combine -4b^{2} and 4b^{2} to get 0.

$$-8a^{2}$$

Expand

$$-8a^{2}$$

Solution Steps

Consider \left(a+2b\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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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}.

$$a^{2}-4b^{2}-\left(\left(3a\right)^{2}-\left(2b\right)^{2}\right)$$

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

$$a^{2}-4b^{2}-\left(3^{2}a^{2}-\left(2b\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

The opposite of -4b^{2} is 4b^{2}.

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

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

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

Combine -4b^{2} and 4b^{2} to get 0.

$$-8a^{2}$$