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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Differences of Squares, Perfect Squares and Factoring Fully

3.2 3.1About 70 minutes
Learning goals
  • I can recognize and factor a difference of squares.
  • I can recognize and factor a perfect square trinomial.
  • I can use a strategy to choose a factoring method and factor an expression fully.
  • I can explain why a sum of squares like x² + 9 does not factor.

Words to know

difference of squares
A binomial of the form a² − b². It factors as (a − b)(a + b).
perfect square trinomial
A trinomial of the form a² + 2ab + b² or a² − 2ab + b². It factors as (a + b)² or (a − b)².
sum of squares
A binomial of the form a² + b². It does not factor over the integers.
factor fully
Keep factoring until no factor can be factored further.

Spot the pattern, save the work

In the first lesson of this unit you learned three special products:

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

Read them backwards and you get three factoring shortcuts. If you can recognize the pattern, you can factor expressions like 4x² − 25 or 9x² − 30x + 25 in one step, without any product-sum searching.

Difference of squares

A difference of squares has exactly two terms, both are perfect squares, and they are subtracted:

a² − b² = (a − b)(a + b)

To factor, take the square root of each term. For 4x² − 25: √(4x²) = 2x and √25 = 5, so 4x² − 25 = (2x − 5)(2x + 5).

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

b²removedABaaArea = a² − b²move BA (turned)Ba + ba − bArea = (a + b)(a − b)
Why a² − b² = (a + b)(a − b): remove a b-by-b square from an a-by-a square. Cut the L-shape into pieces A and B, turn A and slide B beside it. The same area becomes a rectangle (a + b) long and (a − b) wide.
Worked example

Example 1: Difference of squares

Factor 4x² − 25.

  1. Two terms, subtracted, both perfect squares: 4x² = (2x)² and 25 = 5².
  2. Use a² − b² = (a − b)(a + b) with a = 2x and b = 5.
  3. Check: (2x − 5)(2x + 5) = 4x² + 10x − 10x − 25 = 4x² − 25 ✓.

(2x − 5)(2x + 5)

Worked example

Example 2: Common factor first, then a difference of squares

Factor 3x² − 75 fully.

  1. 3x² and 75 are not perfect squares, but they share a factor of 3.
  2. 3(x² − 25).
  3. x² − 25 is a difference of squares: (x − 5)(x + 5).

3(x − 5)(x + 5)

Worked example

Example 3: Factor twice

Factor x⁴ − 16 fully.

  1. x⁴ = (x²)² and 16 = 4². So x⁴ − 16 = (x² − 4)(x² + 4).
  2. x² − 4 is another difference of squares: (x − 2)(x + 2).
  3. x² + 4 is a sum of squares: it does not factor.

(x² + 4)(x − 2)(x + 2)

Perfect square trinomials

A trinomial is a perfect square when:

  1. the first and last terms are perfect squares (and positive), and
  2. the middle term is 2 × (square root of first) × (square root of last), with either sign.

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

For 9x² − 30x + 25: √(9x²) = 3x and √25 = 5. Test the middle: 2 × 3x × 5 = 30x ✓ (and it is negative). So 9x² − 30x + 25 = (3x − 5)².

Worked example

Example 4: Perfect square trinomials

Factor x² + 14x + 49 and 4x² + 12xy + 9y².

  1. x² + 14x + 49: √(x²) = x, √49 = 7, and 2(x)(7) = 14x ✓. So (x + 7)².
  2. 4x² + 12xy + 9y²: √(4x²) = 2x, √(9y²) = 3y, and 2(2x)(3y) = 12xy ✓. So (2x + 3y)².
  3. If the middle test fails, the trinomial is not a perfect square, but it may still factor in another way.

(x + 7)² and (2x + 3y)²

Putting it all together: factoring fully

You now know every type of factoring in the Grade 10 course. When you meet a new expression, follow the same plan every time.

1. Take out the greatest common factor2. Count the termsTwo termsDifference of squares?a² − b² = (a − b)(a + b)(a sum a² + b² will not factor)Three termsPerfect square? (a ± b)²x² + bx + c: product-sumax² + bx + c: decompositionFour termsFactor by groupingthen check the brackets3. Can any factor be factored again?Then expand to check.
A factoring strategy. Always start with a common factor, choose a method by the number of terms, and finish by checking whether any factor can be factored again.
Worked example

Example 5: Several steps

Factor 2x³ − 18x fully.

  1. Common factor 2x: 2x(x² − 9).
  2. Two terms inside, both squares, subtracted: x² − 9 = (x − 3)(x + 3).
  3. None of 2x, x − 3 or x + 3 can be factored further.

2x(x − 3)(x + 3)

Hands-on activity

Cut a square into a rectangle

You need: a sheet of paper, a ruler, a pencil and scissors. Ask a grown-up to help with the scissors.

  1. Draw a square 10 cm by 10 cm and cut it out.
  2. In one corner, draw a 4 cm by 4 cm square and cut it off. Set it aside. The L-shape left has area 100 − 16 = 84 cm².
  3. Cut the L-shape into two rectangles: a 10 cm by 6 cm piece and a 4 cm by 6 cm piece.
  4. Place them end to end to make one long rectangle. Measure it.
  5. Write the difference of squares that matches: 10² − 4² = (10 + 4)(10 − 4).

Model answer: the rectangle is 14 cm by 6 cm, and 14 × 6 = 84 cm², the same as 100 − 16. This is the picture in the lesson with a = 10 and b = 4.

Just for fun

The year 2025 was a perfect square: 45² = 2025. The next perfect-square year is 46² = 2116. The gap between them is 91 years, which is 45 + 46, because 46² − 45² = (46 − 45)(46 + 45). Set a reminder?

Your turn

1

Factor x² − 64.

2

Factor 9x² − 49.

3

Factor 25 − 4y².

4

Factor x² − 18x + 81.

5

Factor 16x² + 40x + 25.

6

Factor 5x² − 20 fully.

7

Factor 2x³ + 12x² + 18x fully.

8

Can x² + 36 be factored over the integers? Explain.

More practice: warm-up

1

Factor x² − 1.

2

Factor x² − 100.

3

Factor x² + 10x + 25.

4

Factor 4x² − 9.

5

Is x² + 6x + 4 a perfect square trinomial? Explain.

More practice: on your own

1

Factor 81x² − 16y².

2

Factor 50x² − 2 fully.

3

Factor x⁴ − 81 fully.

4

Factor 3x² − 24x + 48 fully.

5

Use factoring to find 53² − 47² without a calculator.

Challenge

1

Factor (x + 3)² − 16.

2

Find all values of k that make 9x² + kx + 16 a perfect square trinomial.

3

Factor x²y − 4y − 3x² + 12 fully.

Check your understanding

Quiz

1.

Factor 4x² − 25.

2.

Which is a perfect square trinomial?

3.

What can you say about x² + 49?

4.

Factor 2x² − 72 fully.

5.

Factor 9x² − 12x + 4.

6.

What should you always look for first when factoring?

7.

Use factoring to evaluate 101² − 99².

For parents and teachers

Big idea: This lesson completes expectation 3.2 with differences of squares (such as the curriculum example 4x² − 25) and perfect square trinomials, and then puts all the factoring types together into one strategy.

How to help: Ask your teen to name the type before factoring: "common factor? two terms? three terms? four terms?" The flowchart in the lesson is worth copying onto an index card for the rest of the unit. The paper-cutting activity is a strong visual proof that many students remember for years.

Common difficulties:

  • Trying to factor a sum of squares, such as x² + 9.
  • Stopping too early: x⁴ − 16 needs two steps, and 3x² − 75 needs a common factor first.
  • Calling x² + 6x + 9 a perfect square without testing the middle term (it is, but x² + 6x + 4 is not).