Factor Difference Squares

Grade 9 · algebra · 74 practice problems · read aloud

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Factoring Difference of Squares 🧩

What Is It & Why Is It Useful?

The Difference of Squares is a special pattern where you subtract one perfect square from another. It looks like this: a² - b². Recognizing this pattern is a superpower in algebra because it lets you factor expressions quickly and solve equations more easily!

How to Factor a Difference of Squares

Follow these simple steps:

  1. Identify: Check if you have two perfect squares separated by a subtraction sign.
  2. Find a and b: Determine what squared gives you each term. (What is √(first term)? What is √(second term)?)
  3. Write the Factors: Plug a and b into the pattern: (a + b)(a - b).

Worked Examples

Example 1: Factor x² - 9

  • Step 1: Identify: x² is a perfect square. 9 is a perfect square (3²). They are subtracted.
  • Step 2: Find a and b: a = x, b = 3.
  • Step 3: Write factors: (x + 3)(x - 3).

Answer: (x + 3)(x - 3)

Example 2: Factor 4y² - 25

  • Step 1: Identify: 4y² is (2y)². 25 is 5². They are subtracted.
  • Step 2: Find a and b: a = 2y, b = 5.
  • Step 3: Write factors: (2y + 5)(2y - 5).

Answer: (2y + 5)(2y - 5)

Common Mistakes to Avoid 🚫

  • Forgetting it only works with subtraction: a² + b² (a sum of squares) cannot be factored using real numbers!
  • Misidentifying perfect squares: Check if both terms are truly perfect squares. Is 8x² a perfect square? No, because 8 is not a perfect square.
  • Incorrect signs: The pattern is (a + b)(a - b). Don't mix up the plus and minus signs!

Tips & Tricks

  • Memory Aid: Remember the acronym "SOAP" for the signs: Same, Opposite, Always Positive. The first sign is the same as the original, the second is the opposite, and the last term is always positive.
  • Check your work: Use FOIL to multiply your factors back together. You should get the original expression!

How to Practice

Master this skill by trying these practice ideas:

  • Start simple: x² - 16, 9m² - 1
  • Level up: 16p² - 81q², 50y² - 18 (hint: factor out a GCF first!)
  • Create your own problems and solve them, or swap with a friend.

Practice problems

6 of the 74, worked through step by step — try them before opening the answer.

1 x² - 25 = ?

Hint: This expression follows the pattern a² - b², which can be rewritten as a product of two binomials.

Show the answer

Answer: (x + 5)(x - 5)

  1. Recognize that this is a difference of two squares. A difference of squares has the form: a^2 - b^2 = (a + b)(a - b).
  2. Identify a and b. Here, x^2 is a perfect square: a = x. Also, 25 is a perfect square: 25 = 5^2, so b = 5.
  3. Apply the difference of squares formula. a^2 - b^2 = (a + b)(a - b) Substitute a = x and b = 5: x^2 - 25 = (x + 5)(x - 5)
  4. Check by expanding (optional verification). Multiply (x + 5)(x - 5): = x*x + x*(-5) + 5*x + 5*(-5) = x^2 - 5x + 5x - 25 = x^2 - 25 This matches the original expression, confirming the factorization is correct. Final answer: (x + 5)(x - 5)

We start with the expression: x^2 - 25

2 x² - 49 = ?

Hint: This expression follows the pattern a² - b², which can be rewritten as a product of two binomials. Look for perfect squares in the terms.

Show the answer

Answer: (x + 7)(x - 7)

  1. Recognize that this is a difference of two squares. A difference of squares has the form: a^2 - b^2.
  2. Identify a and b. Here, x^2 is (x)^2, and 49 is (7)^2. So a = x, b = 7.
  3. Apply the difference of squares factoring formula: a^2 - b^2 = (a + b)(a - b)
  4. Substitute a = x and b = 7 into the formula: (x)^2 - (7)^2 = (x + 7)(x - 7)
  5. Check by expanding (optional verification): (x + 7)(x - 7) = x*x + x*(-7) + 7*x + 7*(-7) = x^2 - 7x + 7x - 49 = x^2 - 49 This matches the original expression. Final answer: (x + 7)(x - 7)

We start with the expression: x^2 - 49

3 x⁴ - 16 = ?

Hint: This expression can be viewed as a difference of two perfect squares. Look for a pattern where both terms are perfect squares, then apply the factoring formula. Remember that sometimes you may need to apply the difference of squares method more than once to fully factor an expression.

Show the answer

Answer: (x² + 4)(x + 2)(x - 2)

  1. Recognize the expression as a difference of squares** A difference of squares formula is: a² - b² = (a + b)(a - b) Here, x⁴ = (x²)² and 16 = 4². So we can write: x⁴ - 16 = (x²)² - (4)² Apply the formula with a = x², b = 4: = (x² + 4)(x² - 4) --- **
  2. Notice that x² - 4 is also a difference of squares** x² - 4 = (x)² - (2)² = (x + 2)(x - 2) --- **
  3. Substitute back** From
  4. (x² + 4)(x² - 4) Replace x² - 4 with (x + 2)(x - 2): = (x² + 4)(x + 2)(x - 2) --- **
  5. Final check** We cannot factor x² + 4 further over the real numbers (it would require complex numbers). So the factorization is complete. --- **Final Answer:** (x² + 4)(x + 2)(x - 2)

Let's solve step by step. We start with: x⁴ - 16 --- **

4 x⁶ - 64 = ?

Hint: Look for a pattern where both terms are perfect cubes, and apply the appropriate factoring formula.

Show the answer

Answer: (x³ - 8)(x³ + 8)

  1. Recognize that x⁶ is (x³)² and 64 is 8², so we have a difference of squares: (x³)² - 8²
  2. Apply the difference of squares formula: a² - b² = (a - b)(a + b)
  3. Substitute a = x³ and b = 8 to get (x³ - 8)(x³ + 8) The factored form is (x³ - 8)(x³ + 8).

5 x² - 81 = ?

Hint: This expression follows the pattern a² - b² where both terms are perfect squares

Show the answer

Answer: (x - 9)(x + 9)

  1. Identify the pattern as a difference of squares: a² - b²
  2. Recognize that x² is (x)² and 81 is (9)²
  3. Apply the difference of squares formula: a² - b² = (a - b)(a + b)
  4. Substitute x for a and 9 for b: (x - 9)(x + 9)

The answer is (x - 9)(x + 9).

6 x⁴ - 81 = ?

Hint: This expression can be viewed as a difference of two perfect squares. Look for a pattern where both terms are perfect squares.

Show the answer

Answer: (x² + 9)(x + 3)(x - 3)

  1. Recognize that x⁴ - 81 is a difference of squares: (x²)² - (9)²
  2. Apply the difference of squares formula: a² - b² = (a + b)(a - b)
  3. Substitute a = x² and b = 9: (x² + 9)(x² - 9)
  4. Notice that x² - 9 is also a difference of squares: (x)² - (3)²
  5. Apply the difference of squares formula again: (x + 3)(x - 3)
  6. Combine all factors: (x² + 9)(x + 3)(x - 3) The fully factored form is (x² + 9)(x + 3)(x - 3).
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