Solve (2a-5b)(2a+5b)+(6b-3a)(6b+3a)

$$( 2 a - 5 b ) ( 2 a + 5 b ) + ( 6 b - 3 a ) ( 6 b + 3 a )$$

Evaluate

$$11b^{2}-5a^{2}$$

Solution Steps

Consider \left(2a-5b\right)\left(2a+5b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(2a\right)^{2}-\left(5b\right)^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

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

$$2^{2}a^{2}-\left(5b\right)^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-\left(5b\right)^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Expand \left(5b\right)^{2}.

$$4a^{2}-5^{2}b^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Calculate 5 to the power of 2 and get 25.

$$4a^{2}-25b^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Consider \left(6b-3a\right)\left(6b+3a\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4a^{2}-25b^{2}+\left(6b\right)^{2}-\left(3a\right)^{2}$$

Expand \left(6b\right)^{2}.

$$4a^{2}-25b^{2}+6^{2}b^{2}-\left(3a\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$4a^{2}-25b^{2}+36b^{2}-\left(3a\right)^{2}$$

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

$$4a^{2}-25b^{2}+36b^{2}-3^{2}a^{2}$$

Calculate 3 to the power of 2 and get 9.

$$4a^{2}-25b^{2}+36b^{2}-9a^{2}$$

Combine -25b^{2} and 36b^{2} to get 11b^{2}.

$$4a^{2}+11b^{2}-9a^{2}$$

Combine 4a^{2} and -9a^{2} to get -5a^{2}.

$$-5a^{2}+11b^{2}$$

Expand

$$11b^{2}-5a^{2}$$

Solution Steps

Consider \left(2a-5b\right)\left(2a+5b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(2a\right)^{2}-\left(5b\right)^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

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

$$2^{2}a^{2}-\left(5b\right)^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Calculate 2 to the power of 2 and get 4.

$$4a^{2}-\left(5b\right)^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Expand \left(5b\right)^{2}.

$$4a^{2}-5^{2}b^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Calculate 5 to the power of 2 and get 25.

$$4a^{2}-25b^{2}+\left(6b-3a\right)\left(6b+3a\right)$$

Consider \left(6b-3a\right)\left(6b+3a\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4a^{2}-25b^{2}+\left(6b\right)^{2}-\left(3a\right)^{2}$$

Expand \left(6b\right)^{2}.

$$4a^{2}-25b^{2}+6^{2}b^{2}-\left(3a\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$4a^{2}-25b^{2}+36b^{2}-\left(3a\right)^{2}$$

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

$$4a^{2}-25b^{2}+36b^{2}-3^{2}a^{2}$$

Calculate 3 to the power of 2 and get 9.

$$4a^{2}-25b^{2}+36b^{2}-9a^{2}$$

Combine -25b^{2} and 36b^{2} to get 11b^{2}.

$$4a^{2}+11b^{2}-9a^{2}$$

Combine 4a^{2} and -9a^{2} to get -5a^{2}.

$$-5a^{2}+11b^{2}$$