Difference of Squares: A Quick Factoring Trick

Factoring can sometimes look complicated, but one useful algebra pattern can make certain problems much faster. The difference of squares is a simple factoring trick that lets you turn an expression into two binomials in just a few steps.

The Main Idea

When you subtract one perfect square from another perfect square, you can factor the expression using a special pattern.

The Difference of Squares Formula

a² − b² = (a − b)(a + b)

Small-screen tip: If the formula extends beyond a small phone screen, swipe horizontally inside the equation box.

Why does it work?

Multiply the two factors:

(a − b)(a + b) = a² + ab − ab − b² = a² − b²

The middle terms, +ab and −ab, cancel each other. That leaves the difference of the two squares.

How to Use the Trick

1. Check for subtraction. The expression should have a minus sign between two terms.
2. Check that both terms are perfect squares.
3. Find the square roots. These become the two numbers or expressions inside the factors.
4. Write one factor with subtraction.
5. Write the other factor with addition.
6. Check your answer by multiplying the factors back together.

Worked Example 1: x² − 25

Step 1: Identify the squares.

x² is a square, and 25 = 5².

Step 2: Find their square roots.

The square roots are x and 5.

Step 3: Apply the formula.

x² − 5² = (x − 5)(x + 5)

Answer:

x² − 25 = (x − 5)(x + 5)

Worked Example 2: 81 − 16

Step 1: Recognize the perfect squares.

81 = 9² and 16 = 4².

Step 2: Apply the difference of squares formula.

81 − 16 = 9² − 4² = (9 − 4)(9 + 4)

Step 3: Simplify the factors.

(9 − 4)(9 + 4) = 5 × 13 = 65

Answer: 81 − 16 = 65

Worked Example 3: 100 − 49

Recognize that 100 = 10² and 49 = 7².

100 − 49 = 10² − 7² = (10 − 7)(10 + 7)

= 3 × 17 = 51

Answer: 100 − 49 = 51

Worked Example 4: 9x² − 25

Here, 9x² can be written as (3x)², while 25 = 5².

9x² − 25 = (3x)² − 5²

Now apply the formula:

(3x − 5)(3x + 5)

Answer: 9x² − 25 = (3x − 5)(3x + 5)

A Quick Memory Tip

“Squares minus squares = difference times sum.”

Remember: one factor uses −, and the other uses +.

Common Mistakes to Avoid

• Using addition instead of subtraction: The pattern applies to a² − b², not a² + b².
• Forgetting the plus factor: The answer is (a − b)(a + b), not two subtraction factors.
• Misidentifying perfect squares: Make sure both terms can be written as squares.
• Forgetting to simplify: For numerical problems, multiply the resulting factors when appropriate.
• Not checking the answer: Multiplying the factors back together is a quick way to catch sign errors.

When Can You Use the Difference of Squares?

This factoring pattern works when an expression has the form:

A² − B²

Both A and B can be numbers, variables, or algebraic expressions. The important part is that both terms are squares and they are being subtracted.

Quick Comparison

Expression	Difference of Squares?	Reason
x2 − 16	Yes	x2 and 16 are squares.
x2 + 16	No	The operation is addition.
25y2 − 9	Yes	25y2 = (5y)2 and 9 = 32.
x2 − 10	No	10 is not a perfect square.

Mobile table tip: On a small phone screen, swipe horizontally inside the table to see all columns.

Practice Questions

Try these problems before checking the answers.

1. Factor x² − 36.
2. Factor x² − 64.
3. Factor 49x² − 25.
4. Calculate 121 − 81 using the difference of squares.
5. Factor 16y² − 9.
6. Calculate 1000² − 999² using the difference of squares.

Answers

1. x² − 36 = (x − 6)(x + 6)
2. x² − 64 = (x − 8)(x + 8)
3. 49x² − 25 = (7x − 5)(7x + 5)
4. 121 − 81 = (11 − 9)(11 + 9) = 2 × 20 = 40
5. 16y² − 9 = (4y − 3)(4y + 3)
6. 1000² − 999² = (1000 − 999)(1000 + 999) = 1 × 1999 = 1999

Why This Trick Is Useful

The difference of squares is more than a factoring shortcut. It can also make mental calculations much faster. Instead of multiplying or subtracting very large numbers directly, you can look for nearby perfect squares and use the formula.

For example, the calculation 1000² − 999² looks large, but the formula turns it into 1 × 1999, which is easy to calculate.

Final Takeaway

Whenever you see one square minus another square, think of the difference of squares.

a² − b² = (a − b)(a + b)

Memory tip: Difference of squares = difference × sum. Recognize the pattern, find the square roots, use one minus factor and one plus factor, and then check your result.