A quadratic equation is an equation of the form \[ax^2+bx+c=0\] where $a\neq 0$. “Solving” this equation means finding values of $x$ which satisfy the equation. These values of $x$ are also called the roots of the equation.

There are three commonly-used methods of solving quadratic equations:

Factorising Expressions

We see if a quadratic $q(x)$ can be factorised as $(x+r)(x+s)$ by algebraic manipulation, then the solutions of $q(x)=0$ are $x=-r,\;x=-s$.

Note that it is not always possible to factorise a quadratic expression.

The Quadratic Formula

The quadratic formula will always produce the roots of the equation in the same number of operations, but it's often quicker to use the other methods. You can also use the discriminant to determine how many real roots an equation has.

Completing the Square

This is a not only a reliable method for finding the solutions, if they exist, but also yields an easy way of finding the associated quadratic curve's maximum or minimum point.

Video Example

Prof. Robin Johnson solves the quadratic equations $3x^2-2x-1=0$ by factorisation and $3x^2-4x-2=0$ using the quadratic formula.

Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.