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# (a + b + c)^2 Formula

The (a + b + c)^{2} formula is used to find the sum of squares of three numbers without actually calculating the squares. a plus b plus c Whole Square Formula is one of the major algebraic identities. To derive the expansion of (a + b + c)^2 formula we just multiply (a + b + c) by itself to get A plus B plus C Whole Square. Let us learn more about the (a + b + c)^{2} formula along with solved examples.

## What Is (a + b + c)^{2} Formula?

We just read that by multiplying (a + b + c) by itself we can easily derive the (a + b + c)^{2} formula. Let us see the expansion of (a + b + c)^{2} formula.

(a + b + c)^{2} = (a + b + c)(a + b + c)

(a + b + c)^{2 }= a^{2 }+ ab + ac + ab + b^{2 }+ bc + ca + bc + c^{2}

(a + b + c)^{2 }= a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca

(a + b + c)^{2 }= a^{2} + b^{2} + c^{2 } + 2(ab + bc + ca)

Let us see how to use the (a + b + c)^{2} formula in the following section.

## Examples on (a + b + c)^{2} Formula

Let us take a look at a few examples to better understand the A plus B plus C Whole Square Formula

**Example 1: **Find the value of a^{2} + b^{2} + c^{2} if a + b + c = 20 and ab + bc + ca = 20 using (a + b + c)^{2 }formula.

**Solution:**

To find: a^{2} + b^{2} + c^{2}

Given that:

a + b + c = 20

ab + bc + ca = 2

Using the (a + b + c)^{2} formula,

a^{2} + b^{2} + c^{2} = (a + b + c)^{2} - 2(ab + bc + ca)

a^{2} + b^{2} + c^{2} = (20)^{2} - 2(20) = 400 - 40 = 360

**Answer:** a^{2} + b^{2} + c^{2} = 360.

**Example 2: **Find the value of a^{2} + b^{2} + c^{2} 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:**

To find: a^{2} + b^{2} + c^{2}

Given that:

a + b + c = 5 ... (1)

1/a + 1/b + 1/c = 3 ... (2)

abc = 4 ... (3)

Multiplying equations (2) and (3),

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

bc + ca + ab = 12

Using the (a + b + c)^{2 }formula,

a^{2} + b^{2} + c^{2} = (a + b + c)^{2} - 2(ab + bc + ca)

a^{2} + b^{2} + c^{2} = (5)^{2} - 2(12) = 25 - 24 = 1

**Answer:** a^{2} + b^{2} + c^{2} = 1

**Example 3: **Find the value of a^{2} + b^{2} + c^{2} if a + b + c = 200 and ab + bc + ca = 10000 using (a + b + c)^{2 }formula.

**Solution:**

To find: a^{2} + b^{2} + c^{2}

Given that:

a + b + c = 200

ab + bc + ca = 10000

Using the a^{2} + b^{2} + c^{2} formula,

a^{2} + b^{2} + c^{2} = (a + b + c)^{2} - 2(ab + bc + ca)

a^{2} + b^{2} + c^{2} = (200)^{2} - 2(10000) = 40000 - 20000 = 20000

**Answer:** a^{2} + b^{2} + c^{2} = 20000.

## FAQs on (a + b + c)^{2} Formulas

### What Is the Expansion of (a + b + c)^{2} Formula?

(a + b + c)^{2} formula is read as a plus b plus c whole square. Its expansion is expressed as (a + b + c)^{2 }= a^{2} + b^{2} + c^{2 } + 2(ab + bc + ca).

### What Is the a^{2} + b^{2} + c^{2} Formula in Algebra?

The (a + b + c)^{2} formula is one of the important algebraic identities. It is read as a plus b plus c whole square. The (a + b + c)^{2} formula is expressed as (a + b + c)^{2 }= a^{2} + b^{2} + c^{2 } + 2(ab + bc + ca).

### How To Simplify Numbers Using the A plus B plus C Whole Square Formula?

Let us understand the use of the (a + b + c)^{2} formula with the help of the following example.

**Example:** Find the value of (2 + 5 + 3)^{2} using the (a + b + c)^{2} formula.

To find: (2 + 5 + 3)^{2}

Let us assume that a = 2 and b = 5 and c = 3.

We will substitute these in the formula of (a + b + c)^{2}.

(a + b + c)^{2 }= a^{2} + b^{2} + c^{2 } + 2(ab + bc + ca)

= 2^{2} + 5^{2} + 3^{2} + 2 (2*5 + 5*3 + 3*2)

= 4 + 25 + 9 + 2(10 + 15 + 6) = 38 + 62 = 100

**Answer:** (2 + 5 + 3)^{2} = 100

### How To Use the (a + b + c)^{2} Formula Give Steps?

The following steps are followed while using (a + b + c)^{2} formula.

- Firstly observe the pattern of the numbers whether the sum of three numbers have ^2 as power or not.
- Write down the formula of (a + b + c)
^{2}. - (a + b + c)
^{2 }= a^{2}+ b^{2}+ c^{2 }+ 2(ab + bc + ca) - substitute the value of a, b and c in the (a + b + c)
^{2}formula and simplify.

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