(a + b + c)² and a² – b²

Expand (a + b + c)² and use a² – b² to make calculations easy.

"What is (a + b + c)²?" Use what we already know. Let d = b + c. Then (a + d)² = a² + 2ad + d² = a² + 2a(b + c) + (b + c)².

Open it out: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. In words: every square once, every pair twice. As a picture, it is a square of side a + b + c split into a 3 × 3 grid.

Example 9: 119² = (100 + 10 + 9)² = 10000 + 100 + 81 + 2000 + 1800 + 180 = 14161.

Signs come along with their terms. In (3x – 2y + 4z)², the xy term is 2(3x)(–2y) = –12xy. The squares are always positive.

From Grade 8: a² – b² = (a + b)(a – b), so a² = (a + b)(a – b) + b². Śhrīdharāchārya used this to square numbers fast: move up 5 and down 5 to round numbers. 55² = (55 + 5)(55 – 5) + 5² = 60 × 50 + 25 = 3025.

Numbers ending in 5: 35² = 1225, 65² = 4225, 85² = 7225. For n5, the first digits are n × (n + 1), then write 25. (Śhrīdharāchārya's method shows why: 65² = 70 × 60 + 25.)

(a + b + c)²: each square once, each pair of terms twice.

(a + b + c)² and a² − b² as areas.

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