Difference of Squares

1Summary 重點摘要

Learn an important multiplication formula: (a+b)(a-b) = a² - b².

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The difference of squares formulaThe difference of squares formula states (a+b)(a-b) = a² - b²: the sum of two numbers times their difference equals the difference of their squares. It is an identity, true for all a and b, so it can be applied with confidence. Before using it, recognise the pattern: two binomials multiplied together, where one term is identical in both (both are a) and the other differs only in sign (+b and -b). Without this feature the formula does not apply.
2. Step 2: Why the formula worksExpanding explains it: (a+b)(a-b) = a×a - a×b + b×a - b×b = a² - ab + ab - b². The middle terms -ab and +ab cancel each other, leaving a² - b². Carrying out this expansion once by hand helps the formula stick and shows why the middle terms always vanish. Once the principle is understood, the formula can be re-derived at any time instead of memorised blindly, and related proof questions become manageable.
3. Step 3: Expanding with the formulaOn seeing the form (a+b)(a-b), apply the formula directly instead of multiplying term by term. For example, (x+3)(x-3) = x² - 3² = x² - 9, and the worked example gives (x+6)(x-6) = x² - 6² = x² - 36. First identify what a and b are, then square each; if a term carries a coefficient, such as 2x, square the whole bracketed expression, so (2x)² = 4x². Never square only the number and omit the square of x.
4. Step 4: Factorising with the formulaThe formula also works backwards: on seeing a² - b², write it as (a+b)(a-b), which is factorisation. For example, x² - 16 = x² - 4² = (x+4)(x-4), and x² - 9 = (x+3)(x-3). Before factorising, check that both terms are perfect squares, and make a habit of extracting common factors first: 2x² - 8 = 2(x² - 4) = 2(x+2)(x-2). Taking out the common factor usually simplifies the expression and reveals the difference-of-squares structure.
5. Step 5: Common examples with coefficientsWith coefficients: (2x+5)(2x-5) = (2x)² - 5² = 4x² - 25, and in reverse, 9y² - 25 = (3y)² - 5² = (3y+5)(3y-5). Treat 2x as the whole of a and 5 as the whole of b, squaring the coefficient as well, so (2x)² = 4x² and not 2x². A useful drill is to raise the difficulty gradually: first do x² - 25, then 4x² - 25, practising the idea of treating expressions as wholes until harder expressions no longer feel daunting.
6. Step 6: Conditions and common errorsTwo conditions must hold: both terms are perfect squares and they are subtracted. A sum of squares such as x² + 9 cannot be factorised by this formula — a² + b² has no factorisation into two linear factors. Watch also for two classic slips: writing a² + b² instead of a² - b², and forgetting to square, such as writing (x+6)(x-6) as x² - 6. Review each of these three errors one by one before a test; checking them this way makes them especially memorable.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Formula: (a+b)(a-b) = a² - b²
• Works for both expand and factorise
• Must be two squares subtracted
• Middle terms cancel out
• a² + b² cannot use