Grade by Grade
A. Quadratic Functions · Quadratic Expressions and Equations

Differences of Squares, Perfect Squares and Choosing a Strategy

A1.3About 65 minutes
Learning goals
  • I can factor a difference of squares.
  • I can recognize and factor a perfect square trinomial.
  • I can choose a factoring strategy for any quadratic expression and factor fully.

Words to know

difference of squares
An expression a² − b², which factors as (a − b)(a + b).
perfect square trinomial
A trinomial a² ± 2ab + b², which factors as (a ± b)².
sum of squares
An expression a² + b². It does not factor over the real numbers.
factor fully
Keep factoring until no factor can be factored again.

Patterns worth recognizing

Some quadratic expressions have a special shape, and if you spot it you can factor them in one line. The curriculum names two: differences of squares such as 4x² − 25, and perfect square trinomials such as 9x² + 24x + 16. They are the special products from Expanding and Simplifying Quadratic Expressions, run backward.

Difference of squares

a² − b² = (a − b)(a + b)

Use it when an expression has two terms, both are perfect squares, and they are subtracted.

  • 4x² − 25 = (2x)² − 5² = (2x − 5)(2x + 5)
  • 49 − x² = 7² − x² = (7 − x)(7 + x)
  • 2x² − 50 = 2(x² − 25) = 2(x − 5)(x + 5) (common factor first!)

A sum of squares, such as x² + 25, does not factor over the real numbers. (Try it: no two numbers have product +25 and sum 0.)

b²aaa² − b²bmoveabwidth a + ba − b(a + b)(a − b)
Why a² − b² = (a + b)(a − b): remove a b-by-b square from an a-by-a square, then move the top piece to the end. The leftover area becomes a rectangle a + b wide and a − b tall.

Perfect square trinomials

a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)²

Test a trinomial in three quick checks:

  1. Is the first term a perfect square? (9x² = (3x)²)
  2. Is the last term a perfect square, and positive? (16 = 4²)
  3. Is the middle term twice the product of the square roots? (2 × 3x × 4 = 24x ✓)

If all three pass, 9x² + 24x + 16 = (3x + 4)². The sign inside the bracket matches the sign of the middle term.

Worked example

Example 1: Special products

Factor fully: (a) 9x² + 24x + 16 (b) x² − 14x + 49 (c) 8x² − 18

  1. (a) 9x² = (3x)², 16 = 4², and 2(3x)(4) = 24x. Perfect square: (3x + 4)².
  2. (b) x² = (x)², 49 = 7², and 2(x)(7) = 14x, with a minus sign. (x − 7)².
  3. (c) Common factor 2: 2(4x² − 9). Then 4x² − 9 = (2x)² − 3² is a difference of squares: 2(2x − 3)(2x + 3).
  4. Check (a) by expanding: (3x + 4)² = 9x² + 24x + 16 ✓.

(a) (3x + 4)² (b) (x − 7)² (c) 2(2x − 3)(2x + 3)

Choosing a factoring strategy1. GCF?take it out first2. Two terms?a² − b² factors3. Three terms?square? a = 1? a ≠ 1?4. Checkfully factored?
Use this order every time. Step 3: check for a perfect square, then use product and sum (a = 1) or decomposition (a ≠ 1). Finally make sure no factor can be factored again.
Worked example

Example 2: Choosing the strategy

Factor fully, if possible: (a) 3x² − 27 (b) 5x² + 30x + 45 (c) x² + 16 (d) 6x² − x − 15

  1. (a) GCF 3: 3(x² − 9). Two terms, difference of squares: 3(x − 3)(x + 3).
  2. (b) GCF 5: 5(x² + 6x + 9). Perfect square (9 = 3², 2(x)(3) = 6x): 5(x + 3)².
  3. (c) Two terms, but it is a sum of squares. No GCF. Does not factor.
  4. (d) No GCF; three terms; not a perfect square (6 and −15 are not squares). Decomposition: ac = −90, sum −1: −10 and 9. 6x² − 10x + 9x − 15 = 2x(3x − 5) + 3(3x − 5) = (3x − 5)(2x + 3).

(a) 3(x − 3)(x + 3) (b) 5(x + 3)² (c) does not factor (d) (3x − 5)(2x + 3)

Worked example

Example 3: A difference of squares in disguise

Factor (x + 3)² − 16.

  1. This is A² − B² with A = (x + 3) and B = 4.
  2. A² − B² = (A − B)(A + B) = (x + 3 − 4)(x + 3 + 4).
  3. Simplify each bracket: (x − 1)(x + 7).
  4. Check: expanding the original gives x² + 6x + 9 − 16 = x² + 6x − 7, and (x − 1)(x + 7) = x² + 6x − 7 ✓.

(x − 1)(x + 7)

Watch out

  • x² + 25 is not (x + 5)²: that would be x² + 10x + 25.
  • A perfect square trinomial must have a positive constant term: x² − 6x − 9 is not one.
  • 4x² − 25 is not (4x − 5)(4x + 5): the square root of 4x² is 2x.
  • "Factor fully" means checking each factor again: 2(4x² − 9) is not finished; it is 2(2x − 3)(2x + 3).
Just for fun

Why couldn't x² + 25 get into the factoring party?

It's a sum of squares: it simply has no factors to bring.

Hands-on activity

Cut a difference of squares

You need: a sheet of paper, a ruler, a pencil and scissors (take care with scissors).

  1. Draw and cut out a square 12 cm by 12 cm. This is a² with a = 12.
  2. From one corner, cut away a square 5 cm by 5 cm (b = 5). The leftover piece has area a² − b² = 144 − 25 = 119 cm².
  3. Cut the leftover L-shape into two rectangles with one straight cut, as in the diagram: a 12 cm by 7 cm piece and a 7 cm by 5 cm piece.
  4. Rearrange them into one long rectangle. Measure it.

What to look for: the new rectangle is 17 cm by 7 cm, that is (a + b) by (a − b), and 17 × 7 = 119 cm², the same area as before. So 12² − 5² = (12 + 5)(12 − 5).

Your turn

1

Factor 9x² − 1.

2

Factor x² − 64.

3

Factor 16x² − 40x + 25.

4

Factor x² + 12x + 36.

5

Factor fully: 3x² − 48.

6

Factor fully: 2x² + 20x + 50.

7

Is 4x² + 12x + 25 a perfect square trinomial? Does it factor at all?

More practice: warm-up

1

Factor x² − 9.

2

Factor 25x² − 4.

3

Factor x² + 10x + 25.

4

Factor x² − 2x + 1.

5

Factor 100 − x².

More practice: on your own

1

Factor 0.25x² − 1.

2

Factor fully: 12x² − 75.

3

Factor fully: −x² + 6x − 9.

4

Factor 36x² + 60x + 25.

5

Choose a strategy and factor 4x² + 4x − 15.

Challenge

1

Factor (x + 3)² − 25.

2

Use a difference of squares to work out 51 × 49 without a calculator.

3

Find all values of k that make 9x² + kx + 49 a perfect square trinomial.

Check your understanding

Quiz

1.

Factor 16x² − 9.

2.

Which is a perfect square trinomial?

3.

Which expression cannot be factored over the real numbers?

4.

Factor fully: 2x² − 32.

5.

x² − 10x + 25 factors as:

6.

In what order should you check factoring strategies?

For parents and teachers

Big idea (A1.3): recognizing differences of squares (4x² − 25) and perfect square trinomials (9x² + 24x + 16), and choosing the right strategy for any quadratic expression, always factoring fully.

How to help: give your teen a mixed list from this lesson and the last two and ask them to name the strategy before factoring. Naming the type is half the work.

Common difficulties:

  • Thinking x² + 25 factors. Sums of squares do not.
  • Forgetting the common factor (2x² − 50 is 2(x − 5)(x + 5)).
  • Calling a trinomial a perfect square without checking the middle term.
  • Taking the square root of 4x² as 4x instead of 2x.