Difference of Two Squares (Edexcel IGCSE Maths)

Spec point — 2.2F

Difference of two squares

What is the difference of two squares?

• When a "squared" quantity is subtracted from another "squared" quantity, you get the difference of two squares

  • For example:

    • a² - b²
    • 9² - 5²
    • (x + 1)² - (x - 4)²
    • 4m² - 25n², which is (2m)² - (5n)²

How do I factorise the difference of two squares?

• a²- b² factorises to (a+ b)(a - b)

  • This can be shown by expanding the brackets

    • (a+b)(a−b)=a²−a b+b a−b²=a²−b²

  • The brackets can swap order

    • a²- b² = (a+ b)(a- b) = (a- b)(a + b)
    • (but terms inside a bracket cannot swap order)

• For example, x²−9=(x+3)(x−3)

  • This is the same as (x−3)(x+3)

  • But not the same as (3+x)(3−x)

    • which expands to $9-x^{2}$

How can the difference of two squares be made harder?

• You may find it used with numbers

  • 7² - 3² = (7+3) (7-3) = (10) (4) = 40

• You can use a combination of square numbers and squared variables

  • 4m² - 9n² = (2m)² - (3n)² = (2m + 3n)(2m - 3n)

• You can use other powers which can be written as a difference of two squares

  • r⁸ - t⁶ = (r⁴)² - (t³)² = (r⁴ + t³) (r⁴ - t³)

• You can use the difference of two squares more than once to fully factorise expressions

  • a⁴ - b⁴ = (a²)² - (b²)² = (a² + b²) (a² - b²) = (a² + b²)(a + b)(a - b)

• You may also need to take out a common factor first

  • 2x²−18=2 (x²−9) giving 2 (x+3)(x−3)

    • The 2 comes out in front

Can I use the difference of two squares to expand?

• Using the difference of two squares to expand is quicker than expanding double brackets and collecting like terms

• Brackets of the form ( a + b)( a - b) expand to a² - b²

  • For example (2x+3)(2x−3) expands to $4x^{2}-9$

Exam Hint The difference between two squares is often the "trick" required to complete a harder algebraic question in the exam.

Learning to spot it can help speed up your working.

Worked Example (a) Factorise $9x^{2}-16$.

Answer:

Recognise that $9x^{2}$ and $16$ are both squared terms

Therefore you can factorise using the difference of two squares

Rewrite as a difference of two squared terms

9x²−16=(3x)²−(4)²

Use the rule a²−b²=(a+b)(a−b)

Final answer: $(3x+4)(3x-4)$

(b) Factorise $4r^{2}-t^{4}$.

Answer:

Recognise that $4r^{2}$ and $t^{4}$ are both squared terms

Therefore you can factorise using the difference of two squares

Rewrite as a difference of two squared terms

4r²−t^4=(2r)²−(t²)²

Use the rule a²−b²=(a+b)(a−b)

$(2r+t^{2})(2r-t^{2})$

(c) Factorise $2y^{2}-50$

Answer:

This does not appear to be in the form $a^{2}-b^{2}$

There is a common factor of 2, so take this factor out

2 (y²−25)

You can now see $y^{2}-25$ which has the form $y^{2}-5^{2}$

Use the rule a²−b²=(a+b)(a−b)

y²−25=(y+5)(y−5)

Now multiply this answer by 2 (leaving the 2 on the outside)

2 (y+5)(y−5)