Algebraic Weapons

Study Sheet

Algebraic Weapons

Factorizations and substitutions that crack hard problems

Identities That Do the Work

Tip
Difference of squares, everywhere

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b) — the most-used identity in competition algebra.

In plain terms. A difference of two squares always splits into a product, often collapsing huge arithmetic.

Example. 512−492=(51−49)(51+49)=2⋅100=20051^2 - 49^2 = (51 - 49)(51 + 49) = 2 \cdot 100 = 200 — no squaring needed.

Concept
Symmetric sums via Vieta

By Vieta's formulas, everything symmetric in two roots can be written using x+yx + y and xyxy: for instance x2+y2=(x+y)2−2xyx^2 + y^2 = (x+y)^2 - 2xy.

In plain terms. You rarely need the roots themselves — just their sum and product.

Example. If x+y=6x + y = 6 and xy=5xy = 5, then x2+y2=36−10=26x^2 + y^2 = 36 - 10 = 26.

Reminder — Vieta's formulas:  r1+r2=−ba,r1r2=car_1+r_2=-\frac{b}{a},\qquad r_1 r_2=\frac{c}{a}
Tip
The substitution u=x+1xu = x + \tfrac{1}{x}

From u=x+1xu = x + \tfrac1x: x2+1x2=u2−2x^2 + \tfrac{1}{x^2} = u^2 - 2 and x3+1x3=u3−3ux^3 + \tfrac{1}{x^3} = u^3 - 3u.

In plain terms. Expressions built from xx and 1/x1/x collapse into a polynomial in uu.

Example. If x+1x=3x + \tfrac1x = 3, then x2+1x2=9−2=7x^2 + \tfrac{1}{x^2} = 9 - 2 = 7.