Quadratic roots by factorising
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How to find roots of a quadratic equation by factorising.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to find roots of a quadratic equation by factorising.   
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Algematics software. Click here for instructions.


A quadratic equation looks like this:

ax² + bx + c = 0  (where ‘a’ cannot be zero.)

Solving the equation means finding ‘x’ values that make the equation true. These ‘x’ values are called the roots  of the quadratic.

Quadratic equations can have 0, 1 or two roots.

NOTE: In the complex number system, all quadratic equations have roots, but we will not discuss complex numbers in this article. Roots of quadratics always come in pairs, but when there are two roots that are the same we say that there is only one root. 

This method requires that you can factorise the quadratic expression on the left hand side. This is not always possible, and if not you would have to use one of the other methods.

For a complete explanation of how to factorise quadratic expressions, see the topic: “ Factorise quadratics.

Consider this quadratic expression...

x² - 4x - 5 = 0

When factorised, it looks like this...

(x + 1)(x - 5) = 0

When the quadratic expression is factorised, it is written as the product of two factors,
like this:

pq = 0

This equation is true only if either ‘p’ or ‘q’ is zero.
In the example, ‘p’ is (x+1), and ‘q’ is (x-5).

If ‘p’ is zero, then we have: 0´q = 0 which is true,

and if ‘q’ is zero, then we have: p´0 = 0 which is also true.

For the example, this means that if (x+1) or (x-5) is zero, the product will be zero and the equation will be true. We use this fact to find the roots as follows:

x + 1 = 0  so    x = -1


x - 5 = 0  so    x = 5

 The example quadratic equation has two roots, x = -1 and x = 5.


Download the free support file... We have created an Algematics document containing the completed example from this topic. It also includes practice exercises to improve your skills.

File name:  'Quadratic roots by factorising.alg'   File size: 3kb
Click here
to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.


IMPORTANT: This topic assumes that you know how to enter mathematical formulas into Algematics. Find out how by completing the three simple tutorials in the 'Getting Started' section of the Algematics program 'Help'.

Step 1  Enter the equation to solve

Click  and type the quadratic equation into the 'maths' box in the data entry dialog box.

If the ‘EMPTY’ message is not displayed between the blue buttons, click the  button until the message: ‘INSERT’ appears. If required, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.



   x[2] – 4x – 5




Step 2, Solve...

Keep clicking the  (factorise) button until the equation is factorised, and looks like this:

(x + 1)(x - 5) = 0

If there is a common factor, this will be at the front of the left bracket.

The expressions in brackets must be equated to zero to find the roots.  (See the theory section at the top of this article.)

In this case, x+1 = 0 and x-5 = 0, so the roots are x = -1 and x = 5.

NOTE: If the equation does not factorise, then you will need to use one of the other methods described in the topic: Factorise quadratics.

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