Solving Quadratics

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.

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 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


Sketching Quadratics

It it 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).

quadratics

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.


quadraticsDESMOS 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

QuadraticsQuestions3

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


A-Level Maths

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.