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

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

Evaluate

$$8ac$$

Solution Steps

Square a+b+c.

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

Square b+c-a.

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

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

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

Combine a^{2} and -a^{2} to get 0.

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

Combine 2ab and 2ab to get 4ab.

$$4ab+2ac+b^{2}+2bc+c^{2}+2ac-b^{2}-2bc-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Combine 2ac and 2ac to get 4ac.

$$4ab+4ac+b^{2}+2bc+c^{2}-b^{2}-2bc-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

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

$$4ab+4ac+2bc+c^{2}-2bc-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Combine 2bc and -2bc to get 0.

$$4ab+4ac+c^{2}-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Combine c^{2} and -c^{2} to get 0.

$$4ab+4ac+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Square c+a-b.

$$4ab+4ac+a^{2}-2ab+2ac+b^{2}-2bc+c^{2}-\left(c-b-a\right)^{2}$$

Combine 4ab and -2ab to get 2ab.

$$2ab+4ac+a^{2}+2ac+b^{2}-2bc+c^{2}-\left(c-b-a\right)^{2}$$

Combine 4ac and 2ac to get 6ac.

$$2ab+6ac+a^{2}+b^{2}-2bc+c^{2}-\left(c-b-a\right)^{2}$$

Square c-b-a.

$$2ab+6ac+a^{2}+b^{2}-2bc+c^{2}-\left(a^{2}+2ab-2ac+b^{2}-2bc+c^{2}\right)$$

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

$$2ab+6ac+a^{2}+b^{2}-2bc+c^{2}-a^{2}-2ab+2ac-b^{2}+2bc-c^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$2ab+6ac+b^{2}-2bc+c^{2}-2ab+2ac-b^{2}+2bc-c^{2}$$

Combine 2ab and -2ab to get 0.

$$6ac+b^{2}-2bc+c^{2}+2ac-b^{2}+2bc-c^{2}$$

Combine 6ac and 2ac to get 8ac.

$$8ac+b^{2}-2bc+c^{2}-b^{2}+2bc-c^{2}$$

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

$$8ac-2bc+c^{2}+2bc-c^{2}$$

Combine -2bc and 2bc to get 0.

$$8ac+c^{2}-c^{2}$$

Combine c^{2} and -c^{2} to get 0.

$$8ac$$

Expand

$$8ac$$

Solution Steps

Square a+b+c.

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

Square b+c-a.

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

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

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

Combine a^{2} and -a^{2} to get 0.

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

Combine 2ab and 2ab to get 4ab.

$$4ab+2ac+b^{2}+2bc+c^{2}+2ac-b^{2}-2bc-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Combine 2ac and 2ac to get 4ac.

$$4ab+4ac+b^{2}+2bc+c^{2}-b^{2}-2bc-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

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

$$4ab+4ac+2bc+c^{2}-2bc-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Combine 2bc and -2bc to get 0.

$$4ab+4ac+c^{2}-c^{2}+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Combine c^{2} and -c^{2} to get 0.

$$4ab+4ac+\left(c+a-b\right)^{2}-\left(c-b-a\right)^{2}$$

Square c+a-b.

$$4ab+4ac+a^{2}-2ab+2ac+b^{2}-2bc+c^{2}-\left(c-b-a\right)^{2}$$

Combine 4ab and -2ab to get 2ab.

$$2ab+4ac+a^{2}+2ac+b^{2}-2bc+c^{2}-\left(c-b-a\right)^{2}$$

Combine 4ac and 2ac to get 6ac.

$$2ab+6ac+a^{2}+b^{2}-2bc+c^{2}-\left(c-b-a\right)^{2}$$

Square c-b-a.

$$2ab+6ac+a^{2}+b^{2}-2bc+c^{2}-\left(a^{2}+2ab-2ac+b^{2}-2bc+c^{2}\right)$$

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

$$2ab+6ac+a^{2}+b^{2}-2bc+c^{2}-a^{2}-2ab+2ac-b^{2}+2bc-c^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$2ab+6ac+b^{2}-2bc+c^{2}-2ab+2ac-b^{2}+2bc-c^{2}$$

Combine 2ab and -2ab to get 0.

$$6ac+b^{2}-2bc+c^{2}+2ac-b^{2}+2bc-c^{2}$$

Combine 6ac and 2ac to get 8ac.

$$8ac+b^{2}-2bc+c^{2}-b^{2}+2bc-c^{2}$$

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

$$8ac-2bc+c^{2}+2bc-c^{2}$$

Combine -2bc and 2bc to get 0.

$$8ac+c^{2}-c^{2}$$

Combine c^{2} and -c^{2} to get 0.

$$8ac$$