Foundational Math Toolkit

Polygons, Quadratics & Linear Equations

Property	Formula
Sum of Interior Angles	$$ (n-2) \times 180^\circ $$
Each Interior Angle (Regular Polygon)	$$ \frac{(n-2) \times 180^\circ}{n} $$
Sum of Exterior Angles	Always $$ 360^\circ $$
Number of Diagonals	$$ \frac{n(n-3)}{2} $$

The Formula (Sridharacharya Rule):

For $$ ax^2 + bx + c = 0 $$:

$$ x = \frac{-b \pm \sqrt{D}}{2a} $$

Where $$ D = b^2 – 4ac $$ (Discriminant)

Discriminant (D)	Nature of Roots
$$ D > 0 $$	Real and Distinct (Unequal)
$$ D = 0 $$	Real and Equal (Coincident)
$$ D < 0 $$	No Real Roots (Imaginary/Complex)

For pair: $$ a_1x + b_1y + c_1 = 0 $$ and $$ a_2x + b_2y + c_2 = 0 $$

Condition Ratio	Graphical Meaning	Algebraic Solution
$$ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} $$	Intersecting Lines	Unique Solution (Consistent)
$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} $$	Coincident Lines	Infinite Solutions (Dependent Consistent)
$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} $$	Parallel Lines	No Solution (Inconsistent)

Componendo & Dividendo

Great for simplifying fractions in Trig/Calculus.

$$ \text{If } \frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a+b}{a-b} = \frac{c+d}{c-d} $$

Common Pythagorean Triples

Memorize these for speed in Vectors/3D: