Solve 2(5a+2b)^2-(2a-3b)(2a+3b)

$$2 ( 5 a + 2 b ) ^ { 2 } - ( 2 a - 3 b ) ( 2 a + 3 b )$$

Evaluate

$$46a^{2}+40ab+17b^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(5a+2b\right)^{2}.

$$2\left(25a^{2}+20ab+4b^{2}\right)-\left(2a-3b\right)\left(2a+3b\right)$$

Use the distributive property to multiply 2 by 25a^{2}+20ab+4b^{2}.

$$50a^{2}+40ab+8b^{2}-\left(2a-3b\right)\left(2a+3b\right)$$

Consider \left(2a-3b\right)\left(2a+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}.

$$50a^{2}+40ab+8b^{2}-\left(\left(2a\right)^{2}-\left(3b\right)^{2}\right)$$

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

$$50a^{2}+40ab+8b^{2}-\left(2^{2}a^{2}-\left(3b\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$50a^{2}+40ab+8b^{2}-\left(4a^{2}-\left(3b\right)^{2}\right)$$

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

$$50a^{2}+40ab+8b^{2}-\left(4a^{2}-3^{2}b^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$50a^{2}+40ab+8b^{2}-\left(4a^{2}-9b^{2}\right)$$

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

$$50a^{2}+40ab+8b^{2}-4a^{2}+9b^{2}$$

Combine 50a^{2} and -4a^{2} to get 46a^{2}.

$$46a^{2}+40ab+8b^{2}+9b^{2}$$

Combine 8b^{2} and 9b^{2} to get 17b^{2}.

$$46a^{2}+40ab+17b^{2}$$

Expand

$$46a^{2}+40ab+17b^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(5a+2b\right)^{2}.

$$2\left(25a^{2}+20ab+4b^{2}\right)-\left(2a-3b\right)\left(2a+3b\right)$$

Use the distributive property to multiply 2 by 25a^{2}+20ab+4b^{2}.

$$50a^{2}+40ab+8b^{2}-\left(2a-3b\right)\left(2a+3b\right)$$

Consider \left(2a-3b\right)\left(2a+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}.

$$50a^{2}+40ab+8b^{2}-\left(\left(2a\right)^{2}-\left(3b\right)^{2}\right)$$

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

$$50a^{2}+40ab+8b^{2}-\left(2^{2}a^{2}-\left(3b\right)^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$50a^{2}+40ab+8b^{2}-\left(4a^{2}-\left(3b\right)^{2}\right)$$

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

$$50a^{2}+40ab+8b^{2}-\left(4a^{2}-3^{2}b^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$50a^{2}+40ab+8b^{2}-\left(4a^{2}-9b^{2}\right)$$

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

$$50a^{2}+40ab+8b^{2}-4a^{2}+9b^{2}$$

Combine 50a^{2} and -4a^{2} to get 46a^{2}.

$$46a^{2}+40ab+8b^{2}+9b^{2}$$

Combine 8b^{2} and 9b^{2} to get 17b^{2}.

$$46a^{2}+40ab+17b^{2}$$