Solve (a-b)^2+(b-c)^2+(a+c)^2-(a-b+c)^2

$$( a - b ) ^ { 2 } + ( b - c ) ^ { 2 } + ( a + c ) ^ { 2 } - ( a - b + c ) ^ { 2 }$$

Evaluate

$$a^{2}+b^{2}+c^{2}$$

Solution Steps

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

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

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

$$a^{2}-2ab+b^{2}+b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

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

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+a^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+c^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Combine c^{2} and c^{2} to get 2c^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a-b+c\right)^{2}$$

Square a-b+c.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a^{2}-2ab+2ac+b^{2}-2bc+c^{2}\right)$$

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

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-a^{2}+2ab-2ac-b^{2}+2bc-c^{2}$$

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

$$a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac+2ab-2ac-b^{2}+2bc-c^{2}$$

Combine -2ab and 2ab to get 0.

$$a^{2}+2b^{2}-2bc+2c^{2}+2ac-2ac-b^{2}+2bc-c^{2}$$

Combine 2ac and -2ac to get 0.

$$a^{2}+2b^{2}-2bc+2c^{2}-b^{2}+2bc-c^{2}$$

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

$$a^{2}+b^{2}-2bc+2c^{2}+2bc-c^{2}$$

Combine -2bc and 2bc to get 0.

$$a^{2}+b^{2}+2c^{2}-c^{2}$$

Combine 2c^{2} and -c^{2} to get c^{2}.

$$a^{2}+b^{2}+c^{2}$$

Expand

$$a^{2}+b^{2}+c^{2}$$

Solution Steps

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

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

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

$$a^{2}-2ab+b^{2}+b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+\left(a+c\right)^{2}-\left(a-b+c\right)^{2}$$

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

$$a^{2}-2ab+2b^{2}-2bc+c^{2}+a^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+c^{2}+2ac+c^{2}-\left(a-b+c\right)^{2}$$

Combine c^{2} and c^{2} to get 2c^{2}.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a-b+c\right)^{2}$$

Square a-b+c.

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-\left(a^{2}-2ab+2ac+b^{2}-2bc+c^{2}\right)$$

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

$$2a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac-a^{2}+2ab-2ac-b^{2}+2bc-c^{2}$$

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

$$a^{2}-2ab+2b^{2}-2bc+2c^{2}+2ac+2ab-2ac-b^{2}+2bc-c^{2}$$

Combine -2ab and 2ab to get 0.

$$a^{2}+2b^{2}-2bc+2c^{2}+2ac-2ac-b^{2}+2bc-c^{2}$$

Combine 2ac and -2ac to get 0.

$$a^{2}+2b^{2}-2bc+2c^{2}-b^{2}+2bc-c^{2}$$

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

$$a^{2}+b^{2}-2bc+2c^{2}+2bc-c^{2}$$

Combine -2bc and 2bc to get 0.

$$a^{2}+b^{2}+2c^{2}-c^{2}$$

Combine 2c^{2} and -c^{2} to get c^{2}.

$$a^{2}+b^{2}+c^{2}$$