A quadratic expression is any expression with an x squared term, an x term and a constant. For example, $3x^2-4x+7$ 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.

### 1.  Factorising

Firstly, the simplest method provided that it can be done, is factorising.

Example:

$2x^2-5x-3=0\hspace{5pt}\Rightarrow\hspace{5pt}(2x+1)(x-3)=0\hspace{5pt}\Rightarrow\hspace{5pt} x=-\frac{1}{2}, x=3$.

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 $ax^2+bx+c=0$ then the roots are given by:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Example:

In the following quadratic, a=1, b=3 and c=-3:

$x^2+3x-3=0\hspace{5pt}\Rightarrow\hspace{5pt}x=\frac{-3\pm\sqrt{3^2-4\times 1\times-3}}{2}\hspace{5pt}\Rightarrow\hspace{5pt}x=\frac{-3 +\sqrt{21}}{2}, \frac{-3-\sqrt{21}}{2}$

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

$x^2+6x+5=0\hspace{3pt}\hspace{5pt}\Rightarrow\hspace{5pt}(x+3)^2-4=0\hspace{5pt}\Rightarrow\hspace{5pt} x+3=\pm 2\hspace{3pt}\hspace{5pt}\Rightarrow\hspace{5pt} x=-1, x=-5$

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 $x^2$. For example, $y=(x+3)^2+1$ is the graph of $x^2$ 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 $y=x^2$.

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 Notes in New Window

See more Things to Remember.

Open Quadratics Practice in New Window – solutions are the bottom of the document.