To calculate the sum of two or more squares in an expression, the a^2 + b^2 formula is used. The a2 + b2 formula can be easily derived using the (a+b)2 or (a-b)2 formula. Let us learn these along with a few solved examples in the upcoming sections.
What Is the a^2 + b^2 Formula?
The a^2 + b^2 formula is used to calculate the sum of two or more squares in an expression. Thus, a sum of squares formula or a^2 + b^2 formula can be expressed as:
a2 + b2 = (a +b)2 - 2ab
Also, a2 + b2 = (a - b)2 + 2ab
where, a, b = arbitrary numbers.
Let a and b be the two numbers, the squares of a and b are a2 and b2. The sum of the squares of a and b is a2 + b2. We could obtain a formula using the known algebraic identity (a+b)2 = a2 + b2 + 2ab. On subtracting 2ab from both the sides we can conclude that a2 + b2 = (a +b)2 - 2ab.
Similarly, we can also say that, a2 + b2 = (a - b)2 + 2ab.
Let us have a look at a few solved examples on the a^2 + b^2 formula to understand the concept better.
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Example 1: Using sum of squares formula, find the value of 52 + 62?
To find : value of 52 + 62
Given: a = 5, b = 6
Using sum of squares Formula,
a2 + b2 = (a + b)2 − 2ab
52 + 62 = (5 + 6)2 − 2(5)(6)
= 121 − 2(30)
= 121 − 60
Answer: The value of 52 + 62 is 61.
Example 2 : Verify that the value of x2 + y2 is (x + y)2 - 2xy using a2 + b2 formula.
Solution: To verify x2 + y2 = (x + y)2 - 2xy
Let us use the a2 + b2 formula
a = x, b = y
Using the (a + b)2 formula let us expand the initial terms.
(a + b)2 = a2 + b2 + 2ab
Let us substitute the value of a and b as x and y
(x + y)2 = x2 + y2 + 2xy
On subtracting 2xy from both the sides,
x2 + y2 = (x + y)2 - 2xy
Answer: Hence verified
Example 3: Find the sum of the squares of 10 and 20 directly and using the a^2 + b^2 formula. Verify your answers.
102 + 202 = 100 + 400 = 500
Using the formula a2 + b2 = (a +b)2 -2ab, we get 102 + 202 = (10 + 20)2 - 2 × 10 × 20
= 900- 400
FAQs on a2 + b2 Formula
What Is the Expansion of a2 + b2 Formula?
a2 + b2 formula is known as the sum of squares formula it is read as a square plus b square. Its expansion is expressed as a2 + b2 = (a + b)2 -2ab.
What Is the a2 + b2 Formula in Algebra?
The a2 + b2 formula is one of the important algebraic identities. It is represented by a2 + b2 and is read as a square plus b square. The (a2 + b2) formula is expressed as a2 + b2 = (a +b)2 -2ab.
How To Simplify Numbers Using the a2 + b2 Formula?
Let us understand the use of the a2 + b2 a2 + b2 formula with the help of the following example. Example: Find the value of 202 + 302 using the a2 + b2 formula.
To find: 202 + 302
Let us assume that a = 20 and b = 30.
We will substitute these in the formula of the sum of squares formula that is, a2 + b2
a2 + b2 = (a +b)2 -2ab
202+302 = (20+30)2 - 2(20)(30)
= 2500 - 1200 = 1300 Answer: 202 + 302 = 1300.
How To Use the a2 + b2 Formula Give Steps?
The following steps are followed while using the a2 + b2 formula.
Firstly observe the pattern of the two numbers whether the numbers have ^2 as power or not and in the form of a2 + b2.
Write down the sum of squares formula of a2 + b2 = (a +b)2 -2ab
Substitute the value of a and b in the sum of squares a2 + b2 formula and simplify.