Proof of (a + b + c)2 Formula
Formula for (a + b + c)2 can be proved or expanded in the following two ways:
- Collecting like Terms
- Using Algebraic Identities
Let’s discuss these methods in detail as follows:
Collecting Like Terms
Below we will expand (a + b + c)2 by collecting the like terms.
(a + b + c)2 = (a + b + c) (a + b + c)
⇒ (a + b + c)2 = a (a + b + c) + b (a + b + c) + c (a + b + c)
⇒ (a + b + c)2 = a2 + ab + ac + ba + b2 + bc + ca + cb + c2
From the above expression we will collect all the like terms to get the formula for the (a + b + c)2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
Using Algebraic Identities
Below we will expand (a + b + c)2 using algebraic identity of (x + y)2.
Let p = b + c
⇒ (a + b + c)2 = (a + p)2
By using the algebraic identity (x + y)2 = x2 + y2 + 2xy we get
(a + b + c)2 = a2 + p2 + 2ap
Now putting the value of p in the above expression we get
(a + b + c)2 = a2 + (b + c)2 + 2a (b + c)
Now again using the identity (x + y)2 = x2 + y2 + 2xy expand (b + c)2
(a + b + c)2 = a2 + b2 + c2 + 2bc + 2a (b + c)
⇒ (a + b + c)2 = a2 + b2 + c2 + 2bc + 2ab + 2ac
⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ac)
a plus b plus c Whole Square Formula
a plus b plus c whole square, i.e., (a + b + c)2 formula is one of the important algebraic identities. The formula for (a + b + c)2 is represented as (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca).
In this article, we will explore the (a + b + c)2, a plus b plus c whole square formula, expansion of a plus b plus c whole square, and applications of a plus b plus c whole square. We will also solve some examples on a plus b plus c whole square. Let’s start our learning on the topic (a + b + c)2.