a plus b plus c Whole Square Formula
Last Updated :
26 Aug, 2024
a plus b plus c whole square, i.e., (a + b + c)2 formula is one of the important algebraic identities. It is used very often in Mathematics when solving algebra questions, expanding the squares of a number, etc. It is used to obtain the sum of squares of a number without performing large calculations and increase simplification.
The formula for (a + b + c)2 is represented as:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
In this article, we will explore the (a + b + c)2, a plus b plus c whole square formula, expansion of a plus b plus c whole square, and applications of a plus b plus c whole square. We will also solve some examples on a plus b plus c whole square. Let's start our learning on the topic (a + b + c)2.
What Does (a + b + c) Whole Square Mean?
The (a + b + c)2 means adding three different terms and finding the square of the resultant term. The formula of the (a + b + c) whole square is obtained by the sum of squares of all the three individual terms and the twice of all the product of two terms i.e., ab, bc and ac.
(a + b + c)2 Formula
Below is the formula for (a + b + c)2
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
Proof of (a + b + c)2 Formula
Formula for (a + b + c)2 can be proved or expanded in the following two ways:
Let's discuss these methods in detail as follows:
Collecting Like Terms
Below we will expand (a + b + c)2 by collecting the like terms.
(a + b + c)2 = (a + b + c) (a + b + c)
⇒ (a + b + c)2 = a (a + b + c) + b (a + b + c) + c (a + b + c)
⇒ (a + b + c)2 = a2 + ab + ac + ba + b2 + bc + ca + cb + c2
From the above expression we will collect all the like terms to get the formula for the (a + b + c)2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
Using Algebraic Identities
Below we will expand (a + b + c)2 using algebraic identity of (x + y)2.
Let p = b + c
⇒ (a + b + c)2 = (a + p)2
By using the algebraic identity (x + y)2 = x2 + y2 + 2xy we get
(a + b + c)2 = a2 + p2 + 2ap
Now putting the value of p in the above expression we get
(a + b + c)2 = a2 + (b + c)2 + 2a (b + c)
Now again using the identity (x + y)2 = x2 + y2 + 2xy expand (b + c)2
(a + b + c)2 = a2 + b2 + c2 + 2bc + 2a (b + c)
⇒ (a + b + c)2 = a2 + b2 + c2 + 2bc + 2ab + 2ac
⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ac)
Applications of (a + b + c) Whole Square
There are multiple applications of (a + b + c) whole square. Some of these applications are listed below.
- (a + b + c) whole square is used as an important algebraic identity.
- It is used in physics and engineering to calculate various other formulas.
- It can also be used in financial aspects.
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Examples of (a + b + c) Whole Square
Example 1: Evaluate: (A + 3B + 2C)2.
Solution:
For (A + 3B + 2C)2,
a = A, b = 3B and c = 2C,
Thus, using (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
(A + 3B + 2C)2 = (A)2 + (3B)2 + (2C)2 + 2(A)(3B) + 2(3B)(2C) + 2(A)(2C)
⇒ (A + 3B + 2C)2 = A2 + 9B2 + 4C2 + 6AB + 12BC + 4AC
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:
Given: a + b + c = 5 . . . (i)
1/a + 1/b + 1/c = 3 . . . (ii)
abc = 4 . . . (iii)
To Find: a2 + b2 + c2
Multiply equation (ii) and (iii)
abc (1/a + 1/b + 1/c) = (4)(3)
⇒ bc + ca + ab = 12
Thus, using (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
⇒ a2 + b2 + c2 = (a + b + c)2 - 2(ab + bc + ca)
Thus, a2 + b2 + c2 = (5)2 - 2(12) = 25 - 24 = 1
square,
Conclusion
(a+b+c)2 is an essential mathematical formula to use and is very useful to reduce the amount of calculations and make the problem simplified. It comes handy while solving algebraic equations and deucing them to their simplest forms. It is a useful identity in School level mathematics.
Practice Questions - (a + b + c)2 Formula
Q1. Evaluate: (7x + 2y + 4z)2.
Q2. Find the value of (a + b + c) if a2 + b2 + c2 = 5 and (ab + bc + ac) = 11.
Q3. Find the value of (a2 + b2 + c2) if (a + b + c) = 10 and (ab + bc + ac) = 20.
Q4. Evaluate: (2x + 3 + 19z)2.
Q5. If (a + b + c) = 100 and ab + bc + ca = 34, find the value of the sum of squares of a, b and c.
Q6. If a, b, c are in the ratio of 1:2:3 and ab + bc + ca = -1, then find the value of a, b and c.
Q7. If the sum of squares of a, b and c is 16 and the value of ab + bc + ca = 24, then find the sum of a, b and c.
Q8. Evaluate: (2x - 3u - 5z)2 .
Q9. If (a + b + c) = 200 and the sum of squares of a, b and c is 64, then find the value of ab + bc + ca.
Q10. Find the square of 1003 without multiplying it with itself, use the formula of (a+b+c)2 for simplicity and learning. (Hint: Break 1003 into 1000 + 1 + 2)
FAQs on (a + b + c)2 Formula
What Is the General Formula for (a + b + c) Whole Square?
The general formula for (a + b + c) whole square is given by:
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ac)
How Can I Calculate (a + b + c) Whole Square Quickly?
To calculate (a + b + c) whole square we can use the formula of (a + b + c)2.
What Are the Geometric Interpretations of (a + b + c) Whole Square?
The geometric interpretations of (a + b + c) whole square resembles the area of square with side length equal to the sum of all the three terms.
Can I Apply (a + b + c) Whole Square in Real-Life Problems?
Yes, we can apply (a + b + c) whole square in real life problems such as in financial calculations and in simplifying the complex expressions.
What is the formula for (a-b-c)2 ?
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca
If I have a big number of say 4 digits and want to calculate it's square, can I use the same formula?
Yes. You can convert the single number into sum of digits containing numbers with trailing zeroes to make the calculations easier and then calculate the square of that number.
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