M6 - Advanced

Quadratic formula

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

Solves ax^2 + bx + c = 0. The discriminant b^2 - 4ac tells how many roots.

Difference of two squares

a^2 - b^2 = (a - b)(a + b)

A square subtracted from a square factorises into a product of a difference and a sum.

Vectors

u + v = (x1 + x2, y1 + y2) ; |u| = sqrt(x^2 + y^2)

Add vectors component by component. The length is found with Pythagoras.

Combined probability

independent: P(A and B) = P(A) x P(B)

For independent events, multiply the probabilities.

Quartiles and spread

IQR = Q3 - Q1

Quartiles split ordered data into four parts; the IQR measures spread.

Quadratic functions

y = ax^2 + bx + c

The graph is a parabola. If a > 0 it opens up, if a < 0 it opens down.

SOH CAH TOA

sin = O/H ; cos = A/H ; tan = O/A

In a right triangle these ratios link an angle to two sides.

Exact trigonometric values

sin 30 = 1/2, cos 30 = sqrt3/2, tan 45 = 1

Common angles 0, 30, 45, 60, 90 have exact values worth memorising.

Pythagorean identity

sin^2 x + cos^2 x = 1

This identity holds for every angle.