a2+b2+c2 Formula

What is the a2+b2+c2 Formula

a2+b2+c2 formula is used to find the sum of squares of three numbers 

a2+b2+c2 Formula

a2+b2+c2 formula is used to find the sum of squares of three numbers without actually calculating the squares. a2+b2+c2 formula is one of the major algebraic identities. To derive the expansion of a2+b2+c2 formula evaluate the (a + b + c)2 formula. Let us learn more about the a2+b2+c2 formula

a2+b2+c2 Formula

By multiplying (a + b + c) derive the a2+b2+c2 formula.

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

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

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

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

On subtracting 2bc + 2ca +2ab from both sides

the a2 + c2 + b2 formula is:

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

(or)

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

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

We can also express a2 + b2 + c2 formula as,

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

Solved example of a2 + b2 + c2 formula

Q-1 Find the value of a2 + b2 + c2 if a + b + c = 10 and ab + bc + ca = 5. 

Solution:

In the question it is given that:
a + b + c = 20
ab + bc + ca = 100
Using the a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (10)2 - 2(5) = 100 - 10 = 110

Answer: a2 + b2 + c2 = 110

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a2+b2+c2 Formula

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Frequently Asked Questions on a2+b2+c2 Formula

To use the formula let us take an example in which a=1, b=2 and c=3 and you need to find the value of  a2 + b2 + c2 
To find the value of a2 + b2 + c2 by using a=1, b=2 and c=3

(12 + 22 + 32) using the a2 + b2 + c2 formula.
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
= (1 + 2 + 3)2 - 2(2×1 + 2×3 + 3×1)
= 36 - 22 = 14