a square plus b square formula
a2 + b2 Formula
To calculate the sum of two or more squares in an expression, the a2 + b2 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 a2 + b2 Formula?
The a2 + b2 formula is used to calculate the sum of two or more squares in an expression. Thus, a sum of squares formula or a2 + b2 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.
Lets understand with some solved examples
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