How To Factor Difference of Squares - Algebra - Free Printable

The image shows four expressions that need to be factored using the Difference of Squares rule.

The rule is:
a² – b² = (a – b)(a + b)

Let’s go one by one.

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1. 36x² – 121

- 36x² is (6x)²
- 121 is 11²
→ So, (6x)² – (11)² = (6x – 11)(6x + 11)

✔ Check: (6x – 11)(6x + 11) = 36x² + 66x – 66x – 121 = 36x² – 121 → Correct!

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2. 64x⁴ – 625y⁸

- 64x⁴ = (8x²)²
- 625y⁸ = (25y⁴)²
→ So, (8x²)² – (25y⁴)² = (8x² – 25y⁴)(8x² + 25y⁴)

✔ Check: Multiply back → 64x⁴ – 200x²y⁴ + 200x²y⁴ – 625y⁸ = 64x⁴ – 625y⁸ → Correct!

Note: You might think “Can we factor more?” — but 8x² – 25y⁴ and 8x² + 25y⁴ are not differences of squares themselves (unless you go into radicals, which we don’t in basic algebra). So stop here.

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3. (3x + 5y)² – (4x + 7y)²

This is already in the form A² – B², where:
- A = (3x + 5y)
- B = (4x + 7y)

So apply the rule: (A – B)(A + B)

First, compute A – B:
(3x + 5y) – (4x + 7y) = 3x + 5y – 4x – 7y = –x – 2y

Then, A + B:
(3x + 5y) + (4x + 7y) = 7x + 12y

So answer: (–x – 2y)(7x + 12y)

We can factor out a –1 from the first binomial to make it look nicer:

= –(x + 2y)(7x + 12y)

✔ Check: Let’s multiply (–x – 2y)(7x + 12y):

= –x·7x + –x·12y + –2y·7x + –2y·12y
= –7x² –12xy –14xy –24y²
= –7x² –26xy –24y²

Now check original expression: (3x+5y)² – (4x+7y)²

Expand both:
(9x² + 30xy + 25y²) – (16x² + 56xy + 49y²)
= 9x² + 30xy + 25y² –16x² –56xy –49y²
= –7x² –26xy –24y² → Matches! ✔

So final factored form: –(x + 2y)(7x + 12y)

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4. [(x + 3)² – 16]

This is also a difference of squares:
- (x + 3)² is a square
- 16 is 4²

So: (x + 3)² – 4² = [(x + 3) – 4][(x + 3) + 4]

Simplify each:
- (x + 3 – 4) = (x – 1)
- (x + 3 + 4) = (x + 7)

Answer: (x – 1)(x + 7)

✔ Check: (x – 1)(x + 7) = x² + 7x – x – 7 = x² + 6x – 7

Original: (x + 3)² – 16 = x² + 6x + 9 – 16 = x² + 6x – 7 → Correct!

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Final Answers:

1. 36x² – 121 → (6x – 11)(6x + 11)
2. 64x⁴ – 625y⁸ → (8x² – 25y⁴)(8x² + 25y⁴)
3. (3x + 5y)² – (4x + 7y)² → –(x + 2y)(7x + 12y)
4. [(x + 3)² – 16] → (x – 1)(x + 7)

Final Answer:
(6x – 11)(6x + 11), (8x² – 25y⁴)(8x² + 25y⁴), –(x + 2y)(7x + 12y), (x – 1)(x + 7)