Factorise quadratics
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Factorise quadratic expressions.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to how to factorise quadratic expressions.   
(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.

Theory:

A quadratic expression is one where the highest power on the variable is 2. Here are some examples of quadratic expressions:

1.         x², 

2.         z² + 3z, 

3.         2x² + 3x - 5,

4.         a² - 1, 

5.         p² - 4p - 6

When these quadratic expressions are factorised, they are written as the product of two factors.

This is how to factorise these quadratic expressions:

Example 1:      x²

Write as a product: x´x

Example 2:      z² + 3z

Take z out as a common factor:  z(z + 3)

Example 3:      2x² + 3x - 5

This expression can be factorised if the middle term: 3x,
        is replaced by the equivalent expression: -2x+5x,

 

like this:

                        2x² - 2x + 5x - 5

 After doing this, the first two terms and the last two terms can be factorised:

2x(x - 1) + 5(x - 1)

Now there are only two terms, and there is a common factor: (x - 1), so the expression can be factorised again, like this:

(x - 1)(2x + 5)

Example 4:      a² - 1

This expression can be expressed as: a² - that is a difference of perfect squares, a special pattern that you need to be able to recognise. It can be written immediately as:

(a + 1)(a - 1)

In general, any expression of the form: a² - b²

can then be written immediately as: (a + b)(a - b).

Example 5:      p² - 4p - 6

In this case, the middle term cannot be split to allow factorising as in example 3. You use a technique called 'completing the square' to write the quadratic as a difference of perfect squares, as in example 4 above.

As a difference of perfect squares, the expression becomes:

           

then it factorises like this:

and simplifies to give:

The ‘method’ section below describes how to use Algematics to factorise examples 3, 4 and 5 above.

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:  'Factorise quadratics.alg'   File size: 5kb
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.

Method:

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'.

There are several methods for factorising quadratic expressions, and the Algematics ‘factorise’ and ‘complete the square’ commands can will guide you through the solutions step by step.

Step 1  Enter the data

Click  and type the expression to be factorised 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 necessary, use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.

 

       Maths...

   2x[2] + 3x - 5

 

 

and then click

Step 2  Factorise

Click  (factorise) once. Several things can happen:

1.    If the terms have common factors, like example 2, these will be taken outside the brackets.

2.    If the expression is a difference of perfect squares, like example 4, then it will be factorised immediately.

3.    If you entered a quadratic expression with three terms, like example 3, then Algematics will attempt to split the middle term. If it is successful, you must click  (factorise) twice more to complete the factorisation.

NOTE: To find out how to split the middle terms of quadratic expressions yourself, click here.

4.   If you entered a quadratic expression with three terms and  (factorise) does not change the expression, then you can try completing the square as described in the box below:

 

How to complete the square ...

To complete the square, the x² term of the quadratic expression must be a perfect square.
This is the case in example 5 (using ‘p’ instead of ‘x’):

        p² - 4p - 6

Click  (complete the square). Algematics looks at the first two terms only, giving the equivalent expression:

(p - 2)² - 4 - 6

Click  (Simplify):

(p - 2)² - 10

To make this into the difference of perfect squares, we need to change the number 10 to be something squared.

 is the same as 10 because ‘square root’ and ‘square’ are inverse operations.

Double click on the equation to edit it. Change ‘10’ to ((10)[1:2])[2]

Shortcut buttons in the data entry dialog box make this easy. Select ‘10’ with the mouse, click ,
select (10)[1:2] with the mouse and click .

Click

Now click  (factorise) to factorise as the difference between perfect squares, then  (simplify) to tidy up the result.

 

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