Solve 6(a-2b)^2-(a+b)^2-(a-3b)^2-3b(a-2b)+10ab

$$6 ( a - 2 b ) ^ { 2 } - ( a + b ) ^ { 2 } - ( a - 3 b ) ^ { 2 } - 3 b ( a - 2 b ) + 10 a b$$

Evaluate

$$4a^{2}-13ab+20b^{2}$$

Solution Steps

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

$$6\left(a^{2}-4ab+4b^{2}\right)-\left(a+b\right)^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Use the distributive property to multiply 6 by a^{2}-4ab+4b^{2}.

$$6a^{2}-24ab+24b^{2}-\left(a+b\right)^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

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

$$6a^{2}-24ab+24b^{2}-\left(a^{2}+2ab+b^{2}\right)-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

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

$$6a^{2}-24ab+24b^{2}-a^{2}-2ab-b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Combine 6a^{2} and -a^{2} to get 5a^{2}.

$$5a^{2}-24ab+24b^{2}-2ab-b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Combine -24ab and -2ab to get -26ab.

$$5a^{2}-26ab+24b^{2}-b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Combine 24b^{2} and -b^{2} to get 23b^{2}.

$$5a^{2}-26ab+23b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-3b\right)^{2}.

$$5a^{2}-26ab+23b^{2}-\left(a^{2}-6ab+9b^{2}\right)-3b\left(a-2b\right)+10ab$$

To find the opposite of a^{2}-6ab+9b^{2}, find the opposite of each term.

$$5a^{2}-26ab+23b^{2}-a^{2}+6ab-9b^{2}-3b\left(a-2b\right)+10ab$$

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

$$4a^{2}-26ab+23b^{2}+6ab-9b^{2}-3b\left(a-2b\right)+10ab$$

Combine -26ab and 6ab to get -20ab.

$$4a^{2}-20ab+23b^{2}-9b^{2}-3b\left(a-2b\right)+10ab$$

Combine 23b^{2} and -9b^{2} to get 14b^{2}.

$$4a^{2}-20ab+14b^{2}-3b\left(a-2b\right)+10ab$$

Use the distributive property to multiply -3b by a-2b.

$$4a^{2}-20ab+14b^{2}-3ba+6b^{2}+10ab$$

Combine -20ab and -3ba to get -23ab.

$$4a^{2}-23ab+14b^{2}+6b^{2}+10ab$$

Combine 14b^{2} and 6b^{2} to get 20b^{2}.

$$4a^{2}-23ab+20b^{2}+10ab$$

Combine -23ab and 10ab to get -13ab.

$$4a^{2}-13ab+20b^{2}$$

Expand

$$4a^{2}-13ab+20b^{2}$$

Solution Steps

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

$$6\left(a^{2}-4ab+4b^{2}\right)-\left(a+b\right)^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Use the distributive property to multiply 6 by a^{2}-4ab+4b^{2}.

$$6a^{2}-24ab+24b^{2}-\left(a+b\right)^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

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

$$6a^{2}-24ab+24b^{2}-\left(a^{2}+2ab+b^{2}\right)-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

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

$$6a^{2}-24ab+24b^{2}-a^{2}-2ab-b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Combine 6a^{2} and -a^{2} to get 5a^{2}.

$$5a^{2}-24ab+24b^{2}-2ab-b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Combine -24ab and -2ab to get -26ab.

$$5a^{2}-26ab+24b^{2}-b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Combine 24b^{2} and -b^{2} to get 23b^{2}.

$$5a^{2}-26ab+23b^{2}-\left(a-3b\right)^{2}-3b\left(a-2b\right)+10ab$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-3b\right)^{2}.

$$5a^{2}-26ab+23b^{2}-\left(a^{2}-6ab+9b^{2}\right)-3b\left(a-2b\right)+10ab$$

To find the opposite of a^{2}-6ab+9b^{2}, find the opposite of each term.

$$5a^{2}-26ab+23b^{2}-a^{2}+6ab-9b^{2}-3b\left(a-2b\right)+10ab$$

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

$$4a^{2}-26ab+23b^{2}+6ab-9b^{2}-3b\left(a-2b\right)+10ab$$

Combine -26ab and 6ab to get -20ab.

$$4a^{2}-20ab+23b^{2}-9b^{2}-3b\left(a-2b\right)+10ab$$

Combine 23b^{2} and -9b^{2} to get 14b^{2}.

$$4a^{2}-20ab+14b^{2}-3b\left(a-2b\right)+10ab$$

Use the distributive property to multiply -3b by a-2b.

$$4a^{2}-20ab+14b^{2}-3ba+6b^{2}+10ab$$

Combine -20ab and -3ba to get -23ab.

$$4a^{2}-23ab+14b^{2}+6b^{2}+10ab$$

Combine 14b^{2} and 6b^{2} to get 20b^{2}.

$$4a^{2}-23ab+20b^{2}+10ab$$

Combine -23ab and 10ab to get -13ab.

$$4a^{2}-13ab+20b^{2}$$