Pick any starting number n. We look at three consecutive squares: (n−1)², n², (n+1)². Add the smallest & largest, then subtract twice the middle one.
No matter what n is, the n² terms cancel — leaving exactly 2 every time!
Use (a±b)² to square numbers near round values, e.g. 43² = (40+3)², 29² = (30−1)².
Example: 119² = (100 + 10 + 9)². Try any 3-digit number.
Cut a square of side a, remove a smaller square of side b from the corner, and reassemble the L-shaped remainder into a rectangle of sides (a+b) and (a−b).
Pick a nearby round number b to make the multiplication easy.
Build the rectangle for (x + a)(x + b). The big yellow tile is x², blue strips are x-tiles, red squares are unit tiles.
Enter the coefficients — we'll find a, b such that a+b = p and a×b = q.
A cube of edge (a+b) splits into: 1 cube a³, 3 cuboids a²b, 3 cuboids ab², 1 cube b³.
Example: p³ + 6p²q + 12pq² + 8q³ = (p + 2q)³. Enter coefficients of a²b and ab² terms (assuming a=1 coefficient form a³+3a²b k+3ab²k²+b³k³ pattern) — simpler: just pick a,b below and see the expansion, then reverse-engineer.
Same terms as (a+b)³ but alternating signs.
Given x+y+z, xyz, and x²+y²+z² — find x³+y³+z³ (like Example 15 in the chapter).
We factorise numerator & denominator (as quadratics ax²+bx+c) and cancel common linear factors.