## A square plus b square plus c square formula

The Sum of squares of three numbers can be found by using the formula a2 + b2 + c2 to determine the sum. The formula a2 + b2 + c2 is a fundamental algebraic identity. In this section, we’ll go over some examples and explanations for the formula a2 + b2 + c2.

Proof :

Using (a + b + c) by itself, we can simply get (a2 + b2 + c2) by multiplying it by itself. Here is the formula’s expansion: a2 + b2 + c2.

(a + b + c)2 = (a + b + c)(a + b + c)

= a2 + ab + ac + ab + b2 + bc + ca + bc + c2

= a2 + b2 + c2 + 2ab + 2bc + 2ca

= a2 + b2 + c2 + 2ab + 2bc + 2ca

Subtracting 2ab + 2bc + 2ca from both sides. Therefore, the a2 + b2 + c2 formula is:

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

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

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

So the formula is a2 + b2 + c2 = (a – b – c)2 + 2ab + 2ac – 2bc

## Important points to be noted

It’s an algebraic formula that can factorize numbers.

An algebraic equation displays a scale done on one side of the scale with a number that is also done on the other side.

The numbers are constants in algebraic expressions.

To answer a problem in algebra, we replace letters or alphabets for numbers.

The letters x,y, a,b, etc. can be used to identify the quantities in the equations.

Example :

Find the value of a2 + b2 + c2 if a + b + c = 12 and ab + bc + ca = 10

Solution:

We need to find : a2 + b2 + c2

Given that:

a + b + c = 12

ab + bc + ca = 10

Using the a2 + b2 + c2 formula,

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

a2 + b2 + c2 = (12)2 – 2(10) = 24 – 20 = 4