Applications of Identities

1Summary 重點摘要

Learn how to use the identity formulas we learned to do calculations and factorisation.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Summary of three formulasThree identities must be memorised: (a + b)(a − b) = a² − b², the difference of squares; (a + b)² = a² + 2ab + b², the perfect square of a sum; (a − b)² = a² − 2ab + b², the perfect square of a difference. Note their shapes: the first is two brackets, one all plus and one all minus; the second and third are a bracket times itself, with a middle term of +2ab or −2ab. Memory aid: the difference of squares has no middle term, while a perfect square's middle term is double ab, its sign following the operation inside the bracket.
2. Step 2: How to recognise which to useChoosing the formula: two brackets multiplied, one all plus and one all minus, takes the difference of squares; one bracket times itself takes a perfect square; in reverse, the shape a² − b² factorises as a difference of squares, and a² ± 2ab + b² factorises as a perfect square. Example: x² − 9 is a square minus a square, so the difference of squares gives (x + 3)(x − 3). The key to recognition is the number of terms and the signs, not which letters appear.
3. Step 3: Use formula to calculate fastIdentities make mental arithmetic faster. Example: to find 99², write 99 as 100 − 1 and apply the perfect square of a difference: 99² = (100 − 1)² = 100² − 2 × 100 × 1 + 1² = 10000 − 200 + 1 = 9801, no column working needed. Another: 102 × 98 = (100 + 2)(100 − 2) = 100² − 2² = 10000 − 4 = 9996, two steps by the difference of squares. A common error is writing (100 − 1)² as 100² − 1², dropping the middle term of −200; a perfect square always carries the 2ab term.
4. Step 4: Use formula to factoriseFactorisation is expansion run backwards. Example: x² − 25 = x² − 5² = (x + 5)(x − 5) by the difference of squares; x² + 10x + 25 = x² + 2 × x × 5 + 5² = (x + 5)² by the perfect square of a sum. The steps: decide which identity the shape matches, then slot a and b into place and write the answer. A common error is factorising x² + 10x + 25 correctly as (x + 5)(x + 5) but then slipping into (x + 5)(x − 5), a sign mistake; the check is to expand once and compare.
5. Step 5: Common exam question typesFour exam favourites: expand directly, factorise, speed-calculate with an identity, and evaluate from given conditions. The last kind: given x + y = 5 and xy = 6, find x² + y². Method: from the perfect square identity, x² + y² = (x + y)² − 2xy = 5² − 2 × 6 = 25 − 12 = 13. The key to such questions is rewriting the target expression as a combination of the known conditions, without ever finding x and y separately. Remember the rearrangement x² + y² = (x + y)² − 2xy — it appears constantly.
6. Step 6: Practice is keyIdentities must be memorised, but recognising which one a given expression needs comes only from practice. Aim for at least five questions of each type: a batch of expansions, a batch of factorisations, speed calculations, and evaluate-from-conditions; redo any wrong ones a day or two later, until the shape of an expression instantly suggests the method. A common revision mistake is reading notes without working; mathematics is a doing subject — understanding with the eyes is not doing with the hands. Ten minutes a lesson for a week and the accuracy climbs.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Three formulas must be memorised well
• Recognise form first then use formula
• Using formula is much faster
• Also works for factorisation
• More practice to use flexibly