## a square plus b square formula

### a^{2} + b^{2} Formula

To calculate the sum of two or more squares in an expression, the a^{2} + b^{2} formula is used. The a^{2} + b^{2} 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:

a^{2 }+ b^{2} = (a +b)^{2} – 2ab

Also, a^{2} + b^{2} = (a – b)^{2} + 2ab

where, a, b = arbitrary numbers.

Let a and b be the two numbers, the squares of a and b are a^{2} and b^{2}. The sum of the squares of a and b is a^{2} + b^{2}. We could obtain a formula using the known algebraic identity (a+b)^{2} = a^{2} + b^{2} + 2ab. On subtracting 2ab from both the sides we can conclude that a^{2} + b^{2} = (a +b)^{2} – 2ab.

Similarly, we can also say that, a^{2} + b^{2} = (a – b)^{2} + 2ab.

## Lets understand with some solved examples

**Example 1: Using sum of squares formula, find the value of 5 ^{2 }+ 6^{2}?**

**Solution:**

To find : value of 5^{2} + 6^{2}

Given: a = 5, b = 6

Using sum of squares Formula,

a^{2} + b^{2} = (a + b)^{2 }− 2ab

5^{2} + 6^{2} = (5 + 6)^{2 }− 2(5)(6)

= 121 − 2(30)

= 121 − 60

= 61

**Answer:** The value of 5^{2} + 6^{2} is 61.

**Example 2 : Verify that the value of x ^{2} + y^{2} is (x + y)^{2} – 2xy using a^{2} + b^{2} formula.**

**Solution:** To verify x^{2} + y^{2} = (x + y)^{2 }– 2xy

Let us use the a^{2} + b^{2} formula

a = x, b = y

Using the (a + b)^{2 }formula let us expand the initial terms.

(a + b)^{2} = a^{2 }+ b^{2} + 2ab

Let us substitute the value of a and b as x and y

(x + y)^{2} = x^{2 }+ y^{2} + 2xy

On subtracting 2xy from both the sides,

x^{2} + y^{2} = (x + y)^{2} – 2xy