To solve a quadratic equation, rewrite it in the form \(ax^2+bx+c=0\), then use one of the three methods:

1. Factorising (Link to GCSE)
2. Completing the square
3. Using the quadratic formula

Factorising

For simple quadratics where \(a=1\), find two numbers which add to \(b\) and multiply to \(c\), then put these numbers in double brackets.

For harder quadratics where \(a>1\), start by writing down two brackets, considering the possible coefficients of \(x\). You are still looking for two numbers which multiply to \(c\), but bear in mind that these two numbers will need to multiply by the coefficients of \(x\) before adding to get \(b\).

Once you have the two brackets, equate each one to 0, and solve for \(x\) to get two solutions.

Tip: An element of trial and error is needed for factorising more difficult quadratics. If it looks like there are too many possibilities to try, I would suggest using one of the methods below.

Completing the square

Complete the square by rewriting the equation in the form:

\((x-p)^2-q=0\)

Then solve the equation:

\(\begin{aligned} (x-p)^2&=q \\ x-p&=±\sqrt{q} \\ x&=p±\sqrt{q} \end{aligned}\)

Using the quadratic formula

Identify \(a\), \(b\) and \(c\), and substitute the values into the quadratic formula:

\(x=\dfrac{-b±\sqrt{b^2-4ac}}{2a}\)

Don't forget you can also use a calculator!

Select the equation solver, polynomial degree 2, enter the values for \(a\), \(b\) and \(c\).

Quadratic equations where the unknown is a function

You may come across more complicated "disguised" quadratic equations, where the unknown is not \(x\), but a function, which could be powers of \(x\), trigomometric functions of \(x\), exponential or logarithmic functions of \(x\).

These can be solved by making an appropriate substitution, e.g. \(y=\sin{x}\) and solving the quadratic equation in \(y\), and then unsubstituting to solve \(x\).