Algebraic identities — (a+b)², (a-b)², a²-b², (a+b+c)² with geometric proof

Question

State and prove the four standard algebraic identities using both algebraic expansion and the geometric (area model) method. Then use them to evaluate $103^2$ and $97^2$ without direct multiplication.

(CBSE 8-9 Board pattern — 3-5 marks)

Solution — Step by Step

flowchart TD
    A["Given an expression to simplify"] --> B{"What pattern do you see?"}
    B -- "Sum squared: (x + y)²" --> C["Use Identity 1: a² + 2ab + b²"]
    B -- "Difference squared: (x - y)²" --> D["Use Identity 2: a² - 2ab + b²"]
    B -- "Difference of squares: x² - y²" --> E["Use Identity 3: (a+b)(a-b)"]
    B -- "Three terms squared" --> F["Use Identity 4: expand all pairs"]
    C --> G["Identify a and b, substitute"]
    D --> G
    E --> G
    F --> G
    G --> H["Simplify to get answer"]

Why This Works

These identities are not arbitrary formulas — they come from the distributive property of multiplication. $(a + b)^2 = (a + b)(a + b)$, and when we multiply term by term, we get exactly $a^2 + 2ab + b^2$. The geometric proof confirms this using areas.

The power of identities lies in speed. Computing $103^2$ directly is slow, but splitting it as $(100 + 3)^2$ makes the arithmetic trivial. This is why identities are used in mental maths, competitive exams, and even in computer algorithms.

Alternative Method

For $(a + b + c)^2$, think of it as $((a + b) + c)^2$ and apply Identity 1 twice:

This gives the same result: $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

Common Mistake