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 x, it could by y 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.
Firstly, the simplest method provided that it can be done, is factorising.
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 then the roots are given by:
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.
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 . For example, is the graph of shifted to the left by 3 (x transformation) and then up by 1 (y transformation).
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.
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.