Find the sum of a square- b square +b square-c square +c square- a square =
Answers
Answer:
We just read that by multiplying (a + b + c) by itself we can easily derive the a2 + b2 + c2 formula. Let us see the expansion of a2 + b2 + c2 formula.
(a + b + c)2 = (a + b + c)(a + b + c)
(a + b + c)2 = a2 + ab + ac + ab + b2 + bc + ca + bc + c2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
On subtracting 2ab + 2bc + 2ca from both sides of the above formula, the a2 + b2 + c2 formula is:
a2 + b2 + c2 = (a + b + c)2 - 2 (ab + bc + ca)
(or)
a2 + b2 + c2 = (a + b + c)2 - 2ab - 2bc - 2ca
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
We can also express a2 + b2 + c2 formula as,
a2 + b2 + c2 = (a - b - c)2 + 2ab + 2ac - 2bc
Let us see how to use the a2 + b2 + c2 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 a2 + b2 + c2 .
Example 1: Find the value of a2 + b2 + c2 if a + b + c = 10 and ab + bc + ca = -2.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = 10
ab + bc + ca = -2
Using the a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (10)2 - 2(-2) = 100 + 4 = 104
Answer: a2 + b2 + c2 = 104.
Example 2: Find the value of a2 + b2 + c2 if a + b + c = -3, 1/a + 1/b + 1/c = -2 and abc = 3.
Solution:
To find: a2 + b2 + c2
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 a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (-3)2 - 2(-6) = 9 + 12 = 21
Answer: a2 + b2 + c2 = 21.
Example 3: Find the value of a2 + b2 + c2 if a + b + c = 20 and ab + bc + ca = 100.
Solution:
To find: a2 + b2 + c2
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 = (20)2 - 2(100) = 400 - 200 = 200
Answer: a2 + b2 + c2 = 200.