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Use Algebra to Identify the Roots of a Quadratic

In this worksheet, students will practise using factorisation to find the roots (solutions) for a quadratic graph.

'Use Algebra to Identify the Roots of a Quadratic' worksheet

Key stage:  KS 4

Year:  GCSE

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel,

Curriculum topic:   Algebra, Graphs of Equations and Functions

Curriculum subtopic:   Graphs Graphs of Equations and Functions

Difficulty level:  

Worksheet Overview

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:


Quadratic graphs


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.


A quadratic graph


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



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.

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