Quick formula sheet

A quick-reference collection of the most-used formulae and results. To be consulted while studying and solving exercises.

Algebra

Formula	Name
$(a+b)^2 = a^2+2ab+b^2$	Square of a binomial
$(a-b)(a+b) = a^2-b^2$	Sum times difference
$(a+b)^3 = a^3+3a^2b+3ab^2+b^3$	Cube of a binomial
$a^3+b^3 = (a+b)(a^2-ab+b^2)$	Sum of cubes
$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$	Second-degree equation
$a^{\frac{m}{n}} = \sqrt[n]{a^m}$	Fractional power / radical
$x=\dfrac{\Delta_x}{\Delta},\; y=\dfrac{\Delta_y}{\Delta}$	Cramer’s rule
$x_A+x_B=-\dfrac{b}{a},\; x_Ax_B=\dfrac{c}{a}$	Vieta’s formulae
$\displaystyle\sum_{n=0}^{\infty}q^n = \dfrac{1}{1-q}$, $;	q
$P(B\mid A) = \dfrac{P(A\cap B)}{P(A)}$	Conditional probability
$y = mx+q$, $\; m=\dfrac{n\sum x_iy_i-\sum x_i\sum y_i}{n\sum x_i^2-(\sum x_i)^2}$	Linear regression

Geometry

Result	Context
$\alpha+\beta+\gamma=\pi$	Sum of the interior angles of a triangle
$c^2 = a^2+b^2$	Pythagoras’ theorem
$\dfrac{AH}{CH}=\dfrac{CH}{BH}$	Euclid’s second theorem
$SA\cdot SB = SC\cdot SD$	Intersecting chords theorem
$\dfrac{a+b}{2}\ge\sqrt{ab}$	Arithmetic mean $\ge$ geometric mean
$\alpha_{\text{circ}} = \dfrac{1}{2}\,\alpha_{\text{centro}}$	Inscribed angle
$PA\cong PB$	Tangents from an external point
$R = \dfrac{abc}{4S}$	Circumradius
$r = \dfrac{S}{p}$ ($p$ = semiperimeter)	Inradius
$\ell_{\text{arco}} = \alpha_{\text{rad}}\cdot r$	Arc length
$\dfrac{AK}{KB}=\dfrac{CA}{CB}$	Angle bisector theorem
$A'/A=k^2,\; V'/V=k^3$	Scaling laws (similarity)
Legs $a,a$; hypotenuse $a\sqrt{2}$	$45°$-$45°$-$90°$ triangle

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«Mathematics is not made to be understood, but to be used until it becomes clear.»