Difference of Two Squares (SQA National 5 Maths): Revision Note

Factorising using 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

    • $\left(a+b\right)\left(a-b\right)=a^2-ab+ba-b^2=a^2-b^2$

  • The brackets can be written in either order

    • a² - b² = (a + b)(a - b) = (a - b)(a + b)

    • (but terms inside a bracket cannot swap order)

• For example, $x^2-9=\left(x+3\right)\left(x-3\right)$

  • This is the same as $\left(x-3\right)\left(x+3\right)$

  • But not the same as $\left(3+x\right)\left(3-x\right)$

    • which expands to $9-x^2$

What are some trickier examples of difference of two squares?

• Difference of two squares can be used with:

  • numbers

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

  • A combination of square numbers and squared variables

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

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

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

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

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

  • $2x^2-18=2\left(x^2-9\right)$ giving $2\left(x+3\right)\left(x-3\right)$

    • The 2 comes out in front

Can I use the difference of two squares to expand brackets?

• 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 $\left(2x+3\right)\left(2x-3\right)$ expands to $4x^2-9$

  • Recognising that is quicker than expanding to $4x^2-6x+6x-9$ first and then simplifying