Solving Quadratics
A quadratic expression is any expression with an x squared term, an x term and a constant. For example, 
1. Factorising
Firstly, the simplest method provided that it can be done, is factorising.
Example:
2. Quadratic Formula
If factorising doesn’t work but a quadratic does have roots, the quadratic formula will find them instead. Recall that the discriminant will tell you how many roots a quadratic has. See Discriminants page.
The quadratic formula says that if 
Example:
In the following quadratic, a=1, b=3 and c=-3:
given exactly, i.e. not as a rounded decimal. Rounded to two decimal places using a calculator, the solutions are x=0.79 and x=-3.79.
3. Completing the Square
Alternatively, another infallible method for finding roots if a quadratic can be solved is to complete the square. See Completing the Square page.
Example
Sketching Quadratics
It is worth noting that completing the square is also useful for sketching a quadratic. The reason for this is that, by writing the quadratic in completed square form, we can see the transformations applied to the graph of 
1. Firstly, find the roots using one of the above methods, roots occur when y=0.
2. Then, find the y-intercept, this occurs when x=0.
3. Finally, find the coordinates of the vertex by completing the square and applying transformations to 
See Completing the Square for more details.
DESMOS is a fantastic sketching tool. Click here to try it out. Firstly click the start graphing button and type y=x^2+4x-5 in the bar where the cursor starts. Then try adding more graphs and experimenting with the options. Finally, try exporting your graphs.
Quadratics Practice
Open Quadratics Practice in New Window – solutions are the bottom of the document.
Click here to find Questions by Topic – scroll down to all past QUADRATIC questions.
Are you ready to test your Pure Maths knowledge? If so, visit our Practice Papers page and take StudyWell’s own Pure Maths tests. Alternatively, try the past papers.