Difference of Two Squares is a algebraic expression that takes the form a² − b². This expression is unique because it can be factored into the product of two binomials: (a + b)(a – b). The formula works due to the cancellation of the middle terms when expanding the product of these binomials.

In this article, we will discuss the formula and solved examples for “Difference of Two Squares”.

What is the Difference of Two Squares?

Difference of Two Squares is a fundamental algebraic concept that states:

Difference of two squares is equal to the product of the sum and the difference of the two original numbers.

When you have an expression that involves the subtraction of one square number from another square number, such as a² − b², it can be rewritten in a simpler form using difference of two squares.

Formula for Difference of Two Squares

The formula for the Difference of Two Squares is:

a² − b² = (a + b)(a − b)

Here’s how it works:

• Left Side: a² − b² represents the difference of the squares of two numbers, a and b.
• Right Side: (a + b)(a − b) represents the product of the sum and the difference of the same two numbers, a and b.

Derivation of Difference of Two Square Formula

This formula can be derived and verified through algebraic expansion:

• Start with (a + b)(a – b)
• Apply the distributive property (also known as the FOIL method for binomials):

(a + b)(a – b) = a(a – b) + b(a – b)

⇒ (a + b)(a – b) =a² − ab + ab − b²

⇒ (a + b)(a – b) = a² − b²

Steps to Factor the Difference of Two Squares

To factorize the difference of two squares we can use the following steps:

• Step 1: Identify the Difference of Two Squares.
• Step 2: Express Each Term as a Square.
• Step 3: Apply the Difference of Squares Formula.
• Step 4: Simplify.

Let’s consider some examples for better understanding.

Example: Find the factors of x² − 9.

Solution:

• The expression is x² − 9, which fits the form a² − b².

• Identify
• Express as Squares
  • x² is already a square term.
  • 9 is also a square term: 9 = 3².
• Apply the Formula
  • Using a² − b² = (a + b)(a − b), we get: x² − 9 =(x + 3)(x − 3)
• Simplify
  • No further simplification is needed.

Read More,

• Algebra
• Basics of Algebra
• Algebraic Expression

Solved Examples of Factoring Using Difference of Two Squares

Example 1: Find factors of expression 4y² − 25.

Solution:

The expression is 4y² − 25, which fits the form a² − b².

4y² can be rewritten as (2y)²

25 can be rewritten as 5²

Using a² − b² = (a + b)(a − b), we get: 4y² − 5² =(2y + 5)(2y − 5)

Example 2: Find factors of expression 9a⁴ − 16b².

Solution:

The expression is 9a⁴ − 16b², which fits the form a² − b².

9a⁴ can be rewritten as (3a²)²

16b² can be rewritten as (4b)²

Using a² − b² = (a + b)(a − b), we get: (3a²)² − (4b)² =(3a²+ 4b)(3a² −4b)

Example 3: Find factors of expression 49x² − 64

Solution:

The expression is 49x² − 64, which fits the form a² − b².

49x² can be rewritten as (7x)²

64 can be rewritten as 8²

Using a² − b² = (a + b)(a − b), we get: (7x)² – 8² = (7x + 8)(7x – 8)

Example 4: Find factors of expression 16y⁴ − 81

Solution:

The expression is 16y⁴ − 81, which fits the form a² − b².

16y⁴ can be rewritten as (4y²)²

81 can be rewritten as 9²

Using a² − b² = (a + b)(a − b), we get: (4y²)² – 9² = (4y² + 9)(4y² – 9)

Example 5: Factor 100 − 25z²

Solution:

The expression is 100 − 25z², which fits the form a² − b².

100 can be rewritten as 10²

25z² can be rewritten as 5z²

Using a² − b² = (a + b)(a − b), we get: 10² – 5z² = (10 + 5z)(10 – 5z)

Practice Problems on Difference of Two Squares

Find the factors of the following:

• 64x² − 49
• 121 − 100y²
• 36a² − 16b²
• 169 − 144m²
• 49z² − 25
• 100p² − 81q²
• 25 − 9r²
• 144y⁴ − 64
• 9x² − 4y²

FAQs on Difference of Two Squares

What is the Difference of Two Squares?

The Difference of Two Squares is an algebraic expression of the form a² − b², which can be factored into (a + b)(a – b).

Why does the Difference of Two Squares formula work?

The formula works due to the distributive property of multiplication over addition and subtraction. When you expand (a + b)(a – b), the middle terms cancel out, leaving a² − b².

How can I identify the Difference of Two Squares in an expression?

Look for two terms that are perfect squares and are subtracted from each other. The expression should fit the pattern a2−b2a^2 – b^2a2−b2.

Can the Difference of Two Squares be applied to non-integer terms?

Yes, the formula can be applied to any terms as long as they are perfect squares. These terms can be integers, fractions, variables, or even more complex expressions.

Can the Difference of Two Squares formula be used for higher degree polynomials?

Yes, the formula can be applied to higher degree polynomials if they can be expressed as the difference of two squares. For example, x⁴ − 16 can be factored as (x²)² − 4².