Solve 3a^2+(2a-5b)(2a+5b)-b(a-3b)+22b^2+ab

$$3 a ^ { 2 } + ( 2 a - 5 b ) ( 2 a + 5 b ) - b ( a - 3 b ) + 22 b ^ { 2 } + a b$$

Evaluate

$$7a^{2}$$

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}.

$$3a^{2}+\left(2a\right)^{2}-\left(5b\right)^{2}-b\left(a-3b\right)+22b^{2}+ab$$

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

$$3a^{2}+2^{2}a^{2}-\left(5b\right)^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Calculate 2 to the power of 2 and get 4.

$$3a^{2}+4a^{2}-\left(5b\right)^{2}-b\left(a-3b\right)+22b^{2}+ab$$

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

$$3a^{2}+4a^{2}-5^{2}b^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Calculate 5 to the power of 2 and get 25.

$$3a^{2}+4a^{2}-25b^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Combine 3a^{2} and 4a^{2} to get 7a^{2}.

$$7a^{2}-25b^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Use the distributive property to multiply b by a-3b.

$$7a^{2}-25b^{2}-\left(ba-3b^{2}\right)+22b^{2}+ab$$

To find the opposite of ba-3b^{2}, find the opposite of each term.

$$7a^{2}-25b^{2}-ba+3b^{2}+22b^{2}+ab$$

Combine -25b^{2} and 3b^{2} to get -22b^{2}.

$$7a^{2}-22b^{2}-ba+22b^{2}+ab$$

Combine -22b^{2} and 22b^{2} to get 0.

$$7a^{2}-ba+ab$$

Combine -ba and ab to get 0.

$$7a^{2}$$

Expand

$$7a^{2}$$

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}.

$$3a^{2}+\left(2a\right)^{2}-\left(5b\right)^{2}-b\left(a-3b\right)+22b^{2}+ab$$

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

$$3a^{2}+2^{2}a^{2}-\left(5b\right)^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Calculate 2 to the power of 2 and get 4.

$$3a^{2}+4a^{2}-\left(5b\right)^{2}-b\left(a-3b\right)+22b^{2}+ab$$

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

$$3a^{2}+4a^{2}-5^{2}b^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Calculate 5 to the power of 2 and get 25.

$$3a^{2}+4a^{2}-25b^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Combine 3a^{2} and 4a^{2} to get 7a^{2}.

$$7a^{2}-25b^{2}-b\left(a-3b\right)+22b^{2}+ab$$

Use the distributive property to multiply b by a-3b.

$$7a^{2}-25b^{2}-\left(ba-3b^{2}\right)+22b^{2}+ab$$

To find the opposite of ba-3b^{2}, find the opposite of each term.

$$7a^{2}-25b^{2}-ba+3b^{2}+22b^{2}+ab$$

Combine -25b^{2} and 3b^{2} to get -22b^{2}.

$$7a^{2}-22b^{2}-ba+22b^{2}+ab$$

Combine -22b^{2} and 22b^{2} to get 0.

$$7a^{2}-ba+ab$$

Combine -ba and ab to get 0.

$$7a^{2}$$