(a + b + c)^2 Formula
The (a + b + c)2 formula is used to find the sum of squares of three numbers without actually calculating the squares. a plus b plus c Whole Square Formula is one of the major algebraic identities. To derive the expansion of (a + b + c)^2 formula we just multiply (a + b + c) by itself to get A plus B plus C Whole Square. Let us learn more about the (a + b + c)2 formula along with solved examples.
What Is (a + b + c)2 Formula?
We just read that by multiplying (a + b + c) by itself we can easily derive the (a + b + c)2 formula. Let us see the expansion of (a + b + c)2 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 + 2(ab + bc + ca)
Let us see how to use the (a + b + c)2 formula in the following section.
Examples on (a + b + c)2 Formula
Let us take a look at a few examples to better understand the A plus B plus C Whole Square Formula
Example 1: Find the value of a2 + b2 + c2 if a + b + c = 20 and ab + bc + ca = 20 using (a + b + c)2 formula.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = 20
ab + bc + ca = 2
Using the (a + b + c)2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (20)2 - 2(20) = 400 - 40 = 360
Answer: a2 + b2 + c2 = 360.
Example 2: Find the value of a2 + b2 + c2 if a + b + c = 5, 1/a + 1/b + 1/c = 3 and abc = 4 using A plus B plus C Whole Square Formula.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = 5 ... (1)
1/a + 1/b + 1/c = 3 ... (2)
abc = 4 ... (3)
Multiplying equations (2) and (3),
abc (1/a + 1/b + 1/c) = (4)(3)
bc + ca + ab = 12
Using the (a + b + c)2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (5)2 - 2(12) = 25 - 24 = 1
Answer: a2 + b2 + c2 = 1
Example 3: Find the value of a2 + b2 + c2 if a + b + c = 200 and ab + bc + ca = 10000 using (a + b + c)2 formula.
Solution:
To find: a2 + b2 + c2
Given that:
a + b + c = 200
ab + bc + ca = 10000
Using the a2 + b2 + c2 formula,
a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
a2 + b2 + c2 = (200)2 - 2(10000) = 40000 - 20000 = 20000
Answer: a2 + b2 + c2 = 20000.
FAQs on (a + b + c)2 Formulas
What Is the Expansion of (a + b + c)2 Formula?
(a + b + c)2 formula is read as a plus b plus c whole square. Its expansion is expressed as (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca).
What Is the a2 + b2 + c2 Formula in Algebra?
The (a + b + c)2 formula is one of the important algebraic identities. It is read as a plus b plus c whole square. The (a + b + c)2 formula is expressed as (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca).
How To Simplify Numbers Using the A plus B plus C Whole Square Formula?
Let us understand the use of the (a + b + c)2 formula with the help of the following example.
Example: Find the value of (2 + 5 + 3)2 using the (a + b + c)2 formula.
To find: (2 + 5 + 3)2
Let us assume that a = 2 and b = 5 and c = 3.
We will substitute these in the formula of (a + b + c)2.
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
= 22 + 52 + 32 + 2 (2*5 + 5*3 + 3*2)
= 4 + 25 + 9 + 2(10 + 15 + 6) = 38 + 62 = 100
Answer: (2 + 5 + 3)2 = 100
How To Use the (a + b + c)2 Formula Give Steps?
The following steps are followed while using (a + b + c)2 formula.
- Firstly observe the pattern of the numbers whether the sum of three numbers have ^2 as power or not.
- Write down the formula of (a + b + c)2.
- (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
- substitute the value of a, b and c in the (a + b + c)2 formula and simplify.
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