Question

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

Answer 1

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(a+b+c)2=a2+b2+c2+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=a2+(−b)2+c2+2a(−b)+2(−b)c+2ac

(a-b+c)^2=a^2+b^2+c^2-2ab-2bc+2ac(a−b+c)2=a2+b2+c2−2ab−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=a2+b2+(−c)2+2ab+2b(−c)+2a(−c)

(a+b-c)^2=a^2+b^2+c^2+2ab-2bc-2ac(a+b−c)2=a2+b2+c2+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=a2+b2+c2+2ab+2bc+2ac+a2+b2+c2−2ab−2bc+2ac+a2+b2+c2+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=3a2+3b2+3c2+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)(a+b+c)2+(a−b+c)2+(a+b−c)2=3(a2+b2+c2)+2(ac+ab−bc)

Answer 2

Step-by-step explanation:

I hope the solution is right........