Question

Simplify: (a+b+c)^2+(a-b+c)^2+(a+b-c)^2

Answer 1

(a+b+c)²+(a-b+c)²+(a+b-c)²
=a²+b²+c²+2ab+2bc+2ca+a²+b²+c²-2ab-2bc+2ca+a²+b²+c²+2ab-2bc-2ca
=3a²+3b²+3c²+2ab-2bc+2ca

Answer 2

Answer:

$(a+b+c)^2+(a-b+c)^2+(a+b-c)^2=3(a^2+b^2+c^2)+2(ac+ab-bc)$

Step-by-step explanation:

Given : Expression $(a+b+c)^2+(a-b+c)^2+(a+b-c)^2$

To find : Simplify the expression ?

Solution :

We know that,

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ac$

Similarly solve the second term,

$(a-b+c)^2=a^2+(-b)^2+c^2+2a(-b)+2(-b)c+2ac$

$(a-b+c)^2=a^2+b^2+c^2-2ab-2bc+2ac$

Similarly solve the third term,

$(a+b+c)^2=a^2+b^2+(-c)^2+2ab+2b(-c)+2a(-c)$

$(a+b-c)^2=a^2+b^2+c^2+2ab-2bc-2ac$

Substitute all in the expression,

$(a+b+c)^2+(a-b+c)^2+(a+b-c)^2=a^2+b^2+c^2+2ab+2bc+2ac+a^2+b^2+c^2-2ab-2bc+2ac+a^2+b^2+c^2+2ab-2bc-2ac$

$(a+b+c)^2+(a-b+c)^2+(a+b-c)^2=3a^2+3b^2+3c^2+2ac+2ab-2bc$

$(a+b+c)^2+(a-b+c)^2+(a+b-c)^2=3(a^2+b^2+c^2)+2(ac+ab-bc)$