Number System Fundamentals • Topic 6 of 7

Algebraic Identities

A small set of identities turns awkward arithmetic into mental math. The key ones are (a+b)² = a² + 2ab + b², (a−b)² = a² − 2ab + b², and the difference of squares a² − b² = (a+b)(a−b). Two more are handy: (a+b)² + (a−b)² = 2(a² + b²) and (a+b)² − (a−b)² = 4ab. The difference-of-squares identity is the workhorse — a product like 99 × 101 becomes (100−1)(100+1) = 10000 − 1. Writing a number as (round number ± small number) lets you square or multiply it in a single step.

✅ Solved examples

1. Compute 99 × 101 quickly.
Write it as (100 − 1)(100 + 1) = 100² − 1² = 10000 − 1 = 9999.
2. Find 1007².
Use (1000 + 7)² = 1000² + 2 × 1000 × 7 + 7² = 1000000 + 14000 + 49 = 1014049.
3. Evaluate 17² − 13².
Use a² − b² = (a+b)(a−b) = (17+13)(17−13) = 30 × 4 = 120.
4. If a + b = 10 and a − b = 4, find ab.
Use (a+b)² − (a−b)² = 4ab. So 4ab = 100 − 16 = 84, giving ab = 21.

✏️ Practice — try these, take hints as needed

1. Find 102².
Write 102 as 100 + 2.
Use (a+b)² = a² + 2ab + b².
10000 + 400 + 4.
10404.
2. Compute 53 × 47.
Notice 53 = 50 + 3 and 47 = 50 − 3.
Use (a+b)(a−b) = a² − b².
2500 − 9.
2491.
3. Evaluate 25² − 15².
Use the difference of squares.
(25 + 15)(25 − 15).
40 × 10.
400.
4. If a + b = 12 and ab = 35, find a² + b².
Use a² + b² = (a+b)² − 2ab.
Substitute 12² and 2 × 35.
144 − 70.
74.
5. Find 996².
Write 996 as 1000 − 4.
Use (a−b)² = a² − 2ab + b².
1000000 − 8000 + 16.
992016.

📝 Topic test — 8 questions

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