The a^{2} + b^{2} + c^{2} formula is used to find the sum of squares of three numbers without actually calculating the squares. It says a^{2} + b^{2} + c^{2} = (a + b + c)^{2} - 2ab - 2bc - 2ca.

a^{2} + b^{2} + c^{2} formula is one of the major algebraic identities. To derive the expansion of a^{2} + b^{2} + c^{2 }formula evaluate the (a + b + c)^{2} formula. Let us learn more about the a^{2} + b^{2} + c^{2} formula along with solved examples.

## What is a^2 + b^2 + c^2 Formula?

**a ^{2} + b^{2} + c^{2} formula **says (a + b + c)

^{2}= a

^{2}+ b

^{2}+ c

^{2}+ 2ab + 2bc + 2ca. It is derived from (a + b + c)2 formula as follows.

We just read that by multiplying (a + b + c) by itself we can easily derive the a^{2} + b^{2} + 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} + 2ab + 2bc + 2ca

On subtracting 2ab + 2bc + 2ca from both sides of the above formula, the a^{2} + b^{2} + c^{2} formula is:

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

(or)

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

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

This is the expansion of a^{2} + b^{2} + c^{2} formula.

Similarly, we can also express a^{2} + b^{2} + c^{2 }formula in one of the following ways:

- a
^{2}+ b^{2}+ c^{2 }= (a - b - c)^{2 }+ 2ab + 2ac - 2bc - a
^{2}+ b^{2}+ c^{2 }= (a - b + c)^{2 }+ 2ab - 2ac + 2bc - a
^{2}+ b^{2}+ c^{2 }= (a + b - c)^{2 }- 2ab + 2ac + 2bc, etc.

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

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## Examples on a2 + b2 + c2 Formula

Let us take a look at a few examples to better understand the formula of a squared plus b squared plus c squared.

**Example 1:** Find the value of a^{2} + b^{2} + c^{2} if a + b + c = 10 and ab + bc + ca = -2.

**Solution:**

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

Given that:

a + b + c = 10

ab + bc + ca = -2

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} = (10)^{2} - 2(-2) = 100 + 4 = 104

**Answer:** ∴ a^{2} + b^{2} + c^{2} = 104.

**Example 2:** Find the value of a^{2} + b^{2} + c^{2} if a + b + c = -3, 1/a + 1/b + 1/c = -2 and abc = 3.

**Solution:**

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

Given that:

a + b + c = -3 ... (1)

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

abc = 3 ... (3)

Multiplying (2) and (3),

abc(1/a + 1/b + 1/c) = (3)(−2)

bc + ca + ab = −6

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} = (-3)^{2} - 2(-6) = 9 + 12 = 21

**Answer:** ∴ a^{2} + b^{2} + c^{2} = 21.

**Example 3:** Find the value of a^{2} + b^{2} + c^{2} if a + b + c = 20 and ab + bc + ca = 100.

**Solution:**

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

Given that:

a + b + c = 20

ab + bc + ca = 100

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} = (20)^{2} - 2(100) = 400 - 200 = 200

**Answer:** ∴ a^{2} + b^{2} + c^{2} = 200.

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

### What is the Expansion of a^2 + b^2 + c^2 Formula?

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

### How to Derive a Square Plus b Square Plus c Square Formula Using a2 + b2 Formula?

We know that a^{2} + b^{2} = (a + b)^{2} - 2ab. Replacing b with b + c on both sides:

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

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

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

### What is the a^2 + b^2 + c^2 Formula in Algebra?

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

### How to Simplify Numbers Using the a^{2} + b^{2} + c^{2} Formula?

Let us understand the use of the a^{2} + b^{2} + c^{2} formula with the help of the following example.**Example:** Find the value of (2^{2} + 5^{2} + 3^{2}) using the a^{2} + b^{2} + c^{2} formula.

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

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

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

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

= (2 + 5 + 3)^{2} - 2(2×5 + 5×3 + 3×2)

= 100 - 62 = 38

**Answer:** (2^{2} + 5^{2} + 3^{2}) = 38

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

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

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