Factorise quadratic expressions
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This topic taken straight from the Algematics help database shows you how you can use Algematics to factorise several types of quadratic expressions.

Theory:

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

  1.  x,
  2. z + 3z,
  3.  2x + 3x - 5,
  4. a - 1,
  5.  p - 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: x

Write as a product: xx

Example 2: z + 3z

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

Example 3: 2x + 3x - 5

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

See the end of this article for an explanation of how to split the middle term and write down the factorised quadratic immediately without further steps.

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: a - 1

This expression can be expressed as: a - 1 which 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:
  a - b
can then be written immediately as:
  (a + b)(a - b).

Example 5: p - 4p - 6

In this case, the middle term cannot be split to allow factorising as in example 3.

You can use Algematics to change this expression to a difference of perfect squares, as in example 4 above: 
  

then factorise like this:
  

which simplifies to:
  

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

Method:

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 Set display options

Click to display the 'Set Colours And Fonts' Dialog Box. Make sure that this check box option is not selected:             o Force signs, eg: 2a

and then click

Step 2 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.

Now type your data into the maths box:

        Maths...

 2x[2] + 3x - 5

and then click

Step 3 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.

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

To complete the square, the x term of the quadratic expression must be a perfect square. This is the case in example 5 (using 'p' instead of 'x')
            p - 4p - 6

Click (complete the square). Algematics looks at the first two terms only, giving the equivalent expression:
         (p
- 2) - 4 - 6

Click (Simplify):
           (p - 2) - 10

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

(The square root of 10) 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.

Go back to step 2 to factorise another quadratic.

Practice:

Use Algematics to factorise these quadratic expressions:

(1) x - 100
(2)
x + 5x + 6
(3)
6x + x - 1
(4)
x + 8x - 4
(5)
x + 2x - 2

[ Answers provided in Algematics help file]

How to split the middle term of a quadratic:

It is not always possible to split the middle term of a quadratic expression, but when this is possible, the method described here allows you to write down the factorised quadratic immediately. We will factorise the quadratic expression: 2x + 3x - 5 to demonstrate the technique.

(a) Split the first term into two factors:
            2x = 2x x

(b) Split the last term into two factors:
            -5 = -1 5

(c) Write the factors like this:
               term 1     term 3
   
             2x        -1
    
             x          5

(d) Multiply the factors in opposite corners and add the two answers, these may be the two factors for splitting the middle term:
             2x 5 =
10x
             x
-1 = -x
If they add to give the middle term ('3x' in this case), then the split is correct.

10x - x = 9x
We have not chosen the correct factors.

TRY AGAIN ... BACK TO STEP (c)

(c) Try different term 3 factors:
               term 1     term 3
   
             2x         5
    
             x        -1

(d) Multiply the factors in opposite corners:
             2x -1 = -2x
               x   5  =  
5x

See if they add to give the middle term:
             5x - 2x = 3x

Yes, so we have chosen the correct factors, and the quadratic expression can be written ready to be factorised, like this:
             2x - 2x + 5x - 5

The two top row factors: '2x' and '5' and the two bottom row factors: 'x' and '-1' in step (c) now give the factorised quadratic immediately: 
             (x
- 1)(2x + 5)

 

 

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Last modified: May 01, 2003