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.

What Does (a + b + c) Whole Square Mean?

The (a + b + c)² means adding three different terms and finding the square of the resultant term. The formula of the (a + b + c) whole square is obtained by the sum of squares of all the three individual terms and the twice of all the product of two terms i.e., ab, bc and ac.

(a + b + c)² Formula

Below is the formula for (a + b + c)²

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

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)

Applications of (a + b + c) Whole Square

There are multiple applications of (a + b + c) whole square. Some of these applications are listed below.

• (a + b + c) whole square is used as an important algebraic identity.
• It is used in physics and engineering to calculate various other formulas.
• It can also be used in financial aspects.

Read More About

• Algebraic Identities
• Algebraic Expressions

Examples of (a + b + c) Whole Square

Example 1: Evaluate: (A + 3B + 2C)².

Solution:

For (A + 3B + 2C)²,

a = A, b = 3B and c = 2C,

Thus, using (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(A + 3B + 2C)² = (A)² + (3B)² + (2C)² + 2(A)(3B) + 2(3B)(2C) + 2(A)(2C)

⇒ (A + 3B + 2C)² = A² + 9B² + 4C² + 6AB + 12BC + 4AC

Example 2: Find the value of a² + b² + c² if a + b + c = 5, 1/a + 1/b + 1/c = 3 and abc = 4 using A plus B plus C Whole Square Formula. 

Solution:

Given: a + b + c = 5 . . . (i)

1/a + 1/b + 1/c = 3 . . . (ii)

abc = 4 . . . (iii)

To Find: a² + b² + c²

Multiply equation (ii) and (iii)

abc (1/a + 1/b + 1/c) = (4)(3)

⇒ bc + ca + ab = 12

Thus, using (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

⇒ a² + b² + c² = (a + b + c)² – 2(ab + bc + ca)

Thus, a² + b² + c² = (5)² – 2(12) = 25 – 24 = 1

square,

Practice Questions

Q1. Evaluate: (7x + 2y + 4z)².

Q2. Find the value of (a + b + c) if a² + b² + c² = 5 and (ab + bc + ac) = 11.

Q3. Find the value of (a² + b² + c²) if (a + b + c) = 10 and (ab + bc + ac) = 20.

Frequently Asked Questions (FAQs)

What Is the General Formula for (a + b + c) Whole Square?

The general formula for (a + b + c) whole square is given by:

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

How Can I Calculate (a + b + c) Whole Square Quickly?

To calculate (a + b + c) whole square we can use the formula of (a + b + c)².

What Are the Geometric Interpretations of (a + b + c) Whole Square?

The geometric interpretations of (a + b + c) whole square resembles the area of square with side length equal to the sum of all the three terms.

Can I Apply (a + b + c) Whole Square in Real-Life Problems?

Yes, we can apply (a + b + c) whole square in real life problems such as in financial calculations and in simplifying the complex expressions.