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.
Firstly, the simplest method, provided that it is possible, is factorising.
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.
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 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.
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.