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
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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:
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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.
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