Identities

• What an algebraic identity is — an equation true for all values of the variable(s).
• The three standard Class 8 identities and how to apply them for expansion and factorisation.
• Mental-math shortcuts for squaring numbers near 10, 100, or round bases.
• Real applications in geometry (areas) and quick calculation in commerce.

1. Identity vs equation — An identity holds for every value of the variable (e.g. (x + 3)² = x² + 6x + 9). An equation holds only for specific values (x + 3 = 7 → x = 4).

2. Identity I — (a + b)² = a² + 2ab + b² Example: (3x + 2)² = 9x² + 12x + 4

3. Identity II — (a − b)² = a² − 2ab + b² Example: (5x − 1)² = 25x² − 10x + 1

4. Identity III — (a + b)(a − b) = a² − b² Example: (7x + 4)(7x − 4) = 49x² − 16

5. Numerical applications — 998² = (1000 − 2)² = 1000000 − 4000 + 4 = 996004. 103 × 97 = (100 + 3)(100 − 3) = 10000 − 9 = 9991.

6. Where it shows up — Finding area of a square with side (a + b), difference of areas of two squares, and simplifying products in physics formulas.

Expand (4x − 3y)² using Identity II.

Step 1 — Identify: a = 4x,  b = 3y
Step 2 — a² = (4x)² = 16x²
Step 3 — 2ab = 2(4x)(3y) = 24xy
Step 4 — b² = (3y)² = 9y²
Step 5 — Combine: 16x² − 24xy + 9y²
Answer: 16x² − 24xy + 9y²

Application: A square plot has side (10 + 2) m. Area = (12)² = (10 + 2)² = 100 + 40 + 4 = 144 m² — no long multiplication needed.

• Forgetting the middle term 2ab (writing (a + b)² = a² + b²).
• Sign error in (a − b)²: middle term is −2ab, not +2ab.
• Applying (a + b)² when the expression is (a + b)(a − b) — use Identity III instead.
• Squaring only the first and last terms in three-term expressions.

• Expand (x + 5)².
• Expand (3a − 2b)².
• Evaluate 49² − 51² using Identity III.
• Expand (2x + 7)(2x − 7).

Open the Practice tab for graded questions on Identities.