Quadratics: Factorising, Equations and Graphs

A quadratic expression is a polynomial of degree 2. Its standard form is ax² + bx + c, where 'a', 'b', and 'c' are real numbers and 'a' cannot be zero. The term 'quadratic' comes from 'quadratus', the Latin word for square, as the highest power of the variable is two (squared).

Factorising a quadratic expression means writing it as a product of its factors. Common methods include: finding common factors, using the difference of two squares, recognising perfect squares, and factorising trinomials (x² + bx + c or ax² + bx + c). For trinomials of the form x² + bx + c, we look for two numbers that multiply to 'c' and add to 'b'. For ax² + bx + c (where a ≠ 1), methods like 'splitting the middle term' or the 'cross method' are often used.

A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where 'a', 'b', and 'c' are real numbers and 'a' ≠ 0. Solving a quadratic equation means finding the values of the variable (often 'x') that make the equation true. These values are called the roots or solutions of the equation.

The Null Factor Law states that if the product of two or more factors is zero, then at least one of the factors must be zero. This law is fundamental for solving quadratic equations by factorisation. If (x - p)(x - q) = 0, then x - p = 0 or x - q = 0.

The graph of a quadratic function, y = ax² + bx + c, is a symmetrical curve called a parabola. Key features of a parabola include: the direction it opens (upwards if a > 0, downwards if a < 0), the y-intercept (where x = 0), the x-intercepts (where y = 0, also known as roots or zeros), the axis of symmetry (a vertical line through the vertex), and the vertex (the turning point, which is either a minimum or maximum).