Use Algebra to Identify the Roots of a Quadratic

Quadratics can be dealt with in two ways, algebraically and graphically.

A lot of students see these as two separate topics, however, they do cross over and you can use one technique when dealing with the other.

The significant points of a quadratic graph

All quadratic graphs have the same form, they look something like this:

For all quadratic graphs, there are three things you need to be able to find using an algebraic method - the roots, the y-intercept and the vertex.

In this worksheet, we’re going to look at how we find the roots algebraically.

Example:

How to find the roots of the quadratic y = x² + 3x - 4

This is actually much simpler than you might think.

We need to think what is important about the two points on the graph above that are labelled root 1 and root 2.

They’re on the x–axis which means that the value of y for each of the roots is y = 0

So let’s look at what that gives us:

Step 1: Replace y with zero

This gives the quadratic equation 0 = x² + 3x – 4 which can be rearranged to  x² + 3x – 4 = 0. (Does this look familiar? It should!)

Step 2: Factorise the quadratic

You should be able to do this one quite easily:

x² + 3x – 4 = 0 → (x + 4)(x - 1) = 0

Step 3: Solve the quadratic

For (x + 4)(x - 1) = 0, we can create two equations:

x + 4 = 0 → x = -4

x – 1 = 0 → x = 1

So the roots for the quadratic are x = -4 and x = 1

It's that simple!  Let's have a go at some questions now.