What is the a2+b2+c2 Formula?
The expression a² + b² + c² is a basic algebraic formula that represents the sum of the squares of three variables. In mathematics, squaring a number means multiplying the number by itself. Therefore:
The a2+b2+c2 formula is used to find the sum of the squares of three numbers
(a - b - c)2 + 2ab + 2ac - 2bc = a2 + b2 + c2
a2+b2+c2 Formula
The a2+b2+c2 formula is used to find the sum of squares of three numbers without actually calculating the squares. The a2+b2+c2 formula is one of the major algebraic identities. To derive the expansion of the 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
Importance of the a² + b² + c² Formula
The formula is useful because it appears in many algebraic identities and mathematical problems. It is often used while simplifying expressions, solving equations, and proving identities. Since squared values are always positive or zero, the expression (a² + b² + c²) is also always non-negative.
This formula is also connected with geometry. For example, in coordinate geometry and distance formulas, squares of variables are frequently added together. It is an essential concept for students learning algebraic expansions and factorisation.
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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Conclusion
The a² + b² + c² formula is a fundamental algebraic expression that forms the base for many advanced mathematical concepts. Understanding this formula helps students improve their problem-solving skills and prepares them for topics like algebraic identities, equations, and geometry. Regular practice with examples makes it easier to apply the formula correctly in examinations and daily mathematics practice.
FAQs for 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
The expansion of (a+b+c)2 helps derive the formula:
(a+b+c)2=a2+b2+c2+2(ab+bc+ca)
Rearranging this, we get:
a2+b2+c2=(a+b+c)2−2(ab+bc+ca)
The formula is widely used in:
Physics (kinematics, motion equations)
Engineering (structural analysis, force calculations)
Computer Graphics (3D modeling, transformations)
Machine Learning (distance metrics, vector norms)