Quadratics
Quadratics are mathematical expressions, often represented as axยฒ + bx + c = 0, where ‘a’ is not equal to zero. Quadratic equations describe a specific type of curve in a graphical representation called a parabola. This mathematical construct is essential in physics to depict motion trajectories, in business to calculate profit and loss, and in machine learning algorithms for predictive modeling.
Solving Quadratics
A quadratic expression is any expression with an x squared term, an x term and a constant. For example, 
is a quadratic expression. Note that it doesn’t have to be an 
, it could by 
or any other letter as long as it is the same throughout. Furthermore, there are three ways in which you can solve quadratics – each method requires setting the quadratic to 0 first. See Example 1.
Factorising
Firstly, the simplest method, provided that it is possible, is factorising.
Example:
.
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 
then the roots are given by:
Example: In the following quadratic, 
, 
and 
:
given exactly, i.e. not as a rounded decimal. Rounded to two decimal places using a calculator, the solutions are 
and 
. See Example 2.
Completing the Square
Alternatively, another infallible method for finding roots (if a quadratic is solvable) is to complete the square. See Completing the Square page and Example 3.
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 
(the shape of a quadratic is a known as a parabola). For example, to get 
we shift the graph of by 3 ( transformation) and then up by 1 ( transformation).
1. Firstly, find the roots using one of the above methods, roots occur when 
.
2. Then, find the -intercept, this occurs when 
.
3. Finally, find the coordinates of the vertex by completing the square and applying transformations to 
.
See Completing the Square for more details and check out Example 4.
DESMOS
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.
Examples
Solve the following quadratic equations:
1. 
2. 
Solution:
- Write
as 
and then factorise to 
. The two factors 
and 
could both be zero to give a product of 0 and so the solutions are 
or 
. - First write
as 
and then divide both sides by 2 to give 
. Then factorise to and so the solutions are and 
.
and then factorise to 
. The two factors 
and 
could both be zero to give a product of 0 and so the solutions are 
or 
.
and then divide both sides by 2 to give 
. Then factorise to 
and so the solutions are 
.Solve 
, giving your answers to two decimal places.
Solution:
Since the question says find the solutions to two decimal places, it suggests that we should use the quadratic formula. Selecting 
, 
and 
, the solutions are given by
Solve the equation 
by completing the square. Give exact answers, simplified where appropriate.
Solution:
Taking out a factor of 2 gives the equation:
.
Hence, 
and 
.
Videos
Sketching and finding the discriminant of a quadratic in a completing the square style question.
A real life quadratics question rooted in the context of profit versus selling price.