# a^2 + b^2 Formula

To calculate the sum of two 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.

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

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 (or) 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 }plus 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 prove that, **a ^{2} + b^{2} = (a - b)^{2} + 2ab.**

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## Example on a Square + b Square Formula

**Example 1:** Using the 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
a square plus b square formula 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 square + b square 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

**Answer: Hence verified**

**Example 3:** Find the sum of the squares of 10 and 20 directly and use the a^2 + b^2 formula. Verify your answers.

**Solution: **

10^{2 }+ 20^{2 }= 100 + 400 = 500

Using the formula of a square + b square:

a^{2} + b^{2 }= (a + b)^{2} - 2ab, we get

10^{2 }+ 20^{2 }= (10 + 20)^{2} - 2 × 10 × 20

= 900^{ }- 400

= 500

Thus verified.

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

### What is the Expansion of a^{2} + b^{2} Formula?

**a ^{2 }+ b^{2} formula** is known as the sum of squares formula it is read as a square plus b square. Its expansion is expressed as a

^{2}+ b

^{2 }= (a + b)

^{2}- 2ab.

**☛Also Check:** a^2-b^2 Formula

### How To Simplify Numbers Using the a^{2} + b^{2} Formula?

Let us understand the use of the a^{2} + b^{2} a^{2 }+ b^{2} formula with the help of the following example.

**Example:** Find the value of 20^{2} + 30^{2} using the a^{2} + b^{2} formula.

To find: 20^{2} + 30^{2}

Let us assume that a = 20 and b = 30.

We will substitute these in the formula of the sum of squares formula that is, a^{2} + b^{2}

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

20^{2}+30^{2} = (20+30)^{2} - 2(20)(30)

= 2500 - 1200 = 1300

**Answer:** 20^{2} + 30^{2} = 1300.

### What is the Formula of a Square + b Square in Algebra?

The a square plus b square formula is one of the important algebraic identities. It is represented by a^{2} + b^{2} and is read as a square plus b square. The (a^{2 }+ b^{2}) formula is expressed as a^{2} + b^{2 }= (a + b)^{2} - 2ab.

### How To Use the a^2 + b^2 Formula?

We have two a^{2} + b^{2} formulas and we need to choose the relevant formula depending upon the available information.

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

### What is the Difference Between a Square + b Square and a Square - b Square Formulas?

- a square plus b square formula is: a
^{2}+ b^{2 }= (a + b)^{2}- 2ab (or) (a - b)^{2}+ 2ab - a square minus b square formula is: a
^{2}- b^{2}= (a - b) (a + b)

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