a^2 + b^2 Formula | A Square Plus B Square Formula with Examples

The a² + b² formula represents the sum of the squares of two numbers or variables, a and b. It can be written using either the sum or difference of the two terms:

a² + b² = (a + b)² − 2ab

or

a² + b² = (a − b)² + 2ab

These forms are obtained from the identities for the square of a sum and the square of a difference.

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What Is the a² + b² Formula?

The a² + b² formula is used to find the sum of the squares of two terms when their sum and product, or their difference and product, are known.

The two useful forms are:

a² + b² = (a + b)² − 2ab

a² + b² = (a − b)² + 2ab

The form you use depends on the information given in the question.

For example, if a + b and ab are known, use:

a² + b² = (a + b)² − 2ab

If a − b and ab are known, use:

a² + b² = (a − b)² + 2ab

a Square Plus b Square Formula

The a square plus b square formula is:

a² + b² = (a + b)² − 2ab

It can also be written as:

a² + b² = (a − b)² + 2ab

These are not two different identities. They are two equivalent ways of expressing a² + b².

Why Are There Two Forms?

Start with the identity:

(a + b)² = a² + 2ab + b²

Rearranging gives:

a² + b² = (a + b)² − 2ab

Similarly:

(a − b)² = a² − 2ab + b²

Rearranging gives:

a² + b² = (a − b)² + 2ab

Read More: (a + b)³ Formula

Derivation of the a² + b² Formula

Using (a + b)²

We know:

(a + b)² = a² + 2ab + b²

Move 2ab to the other side:

a² + b² = (a + b)² − 2ab

Therefore:

a² + b² = (a + b)² − 2ab

Using (a − b)²

We know:

(a − b)² = a² − 2ab + b²

Add 2ab to both sides:

a² + b² = (a − b)² + 2ab

Therefore:

a² + b² = (a − b)² + 2ab

How to Find a² + b²

Use these steps:

1. Check whether the question gives a + b or a − b.

2. Identify the value of ab if it is given.

3. Choose the corresponding formula.

4. Substitute the known values.

5. Simplify to find a² + b².

Example 1: If a + b = 10 and ab = 21

Given:

a + b = 10

ab = 21

Use:

a² + b² = (a + b)² − 2ab

Substitute the values:

= 10² − 2(21)

= 100 − 42

= 58

Therefore:

a² + b² = 58

Example 2: Find 15² + 5²

Let:

a = 15, b = 5

Then:

a + b = 20

and

ab = 75

Using:

a² + b² = (a + b)² − 2ab

= 20² − 2(75)

= 400 − 150

= 250

Therefore:

15² + 5² = 250

Example 3: If a − b = 7 and ab = 6

Given:

a − b = 7

ab = 6

Use:

a² + b² = (a − b)² + 2ab

Substitute:

= 7² + 2(6)

= 49 + 12

= 61

Therefore:

a² + b² = 61

Read More: A^2-B^2 Formula

a² + b² Formula Expansion

The expression a² + b² is already the sum of two squares. Unlike (a + b)², it does not contain the middle term 2ab.

Compare:

(a + b)² = a² + 2ab + b²

but

a² + b² = (a + b)² − 2ab

This distinction is important when solving algebraic expressions.

For example, if a = 2 and b = 3:

a² + b² = 2² + 3² = 4 + 9 = 13

But:

(a + b)² = 5² = 25

So:

a² + b² ≠ (a + b)²

Difference Between a² + b² and a² − b²

The GSC data also shows strong demand for a² − b² formula, but it is a different identity.

a² + b²

a² + b² = (a + b)² − 2ab

or

a² + b² = (a − b)² + 2ab

a² − b²

The difference of squares can be factorised as:

a² − b² = (a + b)(a − b)

So, a² + b² and a² − b² should not be treated as the same formula. The difference-of-squares identity is a separate algebraic identity.

a² + b² and (a + b)²

These expressions are often confused.

(a + b)² = a² + 2ab + b²

Therefore:

a² + b² = (a + b)² − 2ab

The term 2ab is the difference between the two expressions.

For example, when a = 3 and b = 2:

a² + b² = 9 + 4 = 13

while:

(a + b)² = 5² = 25

The difference is:

25 − 13 = 12 = 2(3)(2)

Uses of the a² + b² Formula

The formula is useful when a question gives information such as:

• The sum a + b and product ab

• The difference a − b and product ab

• An algebraic expression involving squares

• Values that can be substituted into an algebraic identity

The sum of squares also appears in geometry. For a right triangle, the Pythagorean theorem relates the squares of the two legs to the square of the hypotenuse:

a² + b² = c²

Here, a and b represent the perpendicular sides, while c represents the hypotenuse. CuriousJr helps students understand these parts of a right triangle clearly.

Common Mistakes in a² + b² Formula

1. Confusing a² + b² with (a + b)²

Incorrect:

a² + b² = (a + b)²

Correct:

a² + b² = (a + b)² − 2ab

2. Forgetting 2ab

When using the sum form, remember:

a² + b² = (a + b)² − 2ab

The 2ab term cannot be omitted.

3. Using the wrong sign

For the sum form:

a² + b² = (a + b)² − 2ab

For the difference form:

a² + b² = (a − b)² + 2ab

The signs outside the brackets are different.

4. Confusing a² + b² with a² − b²

The expressions represent different operations:

a² + b² → sum of squares

a² − b² → difference of squares

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