the simplified value of expression (a+b-c)^2+2(a+b-c)(a-b+c)+(a-b+c)^2 is
Answers
Answer:
(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(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(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
=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
+2a(−b)+2(−b)c+2ac
(a-b+c)^2=a^2+b^2+c^2-2ab-2bc+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+2b(−c)+2a(−c)
(a+b-c)^2=a^2+b^2+c^2+2ab-2bc-2ac(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
=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
=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)(a+b+c)
2
+(a−b+c)
2
+(a+b−c)
2
=3(a
2
+b
2
+c
2
)+2(ac+ab−bc)