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
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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Frequently Asked Questions on a2+b2+c2 Formula
. How to use numbers using the formula a2 + b2 + c2 ?
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