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Split the middle term of a quadratic equation in order to factorise it. This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to how to split the middle term of a quadratic equation in order to factorise it.Â Â Â  (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Algematics software. Click here for instructions. Â

Algematics can easily split the middle term of many quadratic expressions in order to factorise them. You must be able to do this step yourself without help from Algematics, and this topic shows you how to do it. 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.

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We will factorise the quadratic expression: 2x² + 3x - 5 to demonstrate the technique. Here are the steps:

(a) Split the first term into two factors with the variable in each one:

2x²Â  =Â  2x ´ x

(b) Split the last term into two factors:

-5Â  =Â  -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:

 term 1 term 3 2x -1 x 5

Â 2x ´ 5 + x ´ -1 = 10x - x
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = 9x

If the two products add to give the middle term then the split is correct. But in this case, the middle term is 3x, not 9x.

Â We have not chosen the correct factors.Â  TRY AGAIN ... BACK TO STEP (c)

(c) Try different term 3 factors, or the same factors in a different arrangement:

 term 1 term 3 2x 5 x -1

(d) Multiply the factors in opposite corners:

 term 1 term 3 2x 5 x -1

See if the two products 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:

2x² - 2x + 5x - 5

The two factors from the top row of the table: ‘2x’ and ‘5’ and the two factors from the bottom row of the table: ‘x’ and ‘-1’ in step (c) now give the factorised quadratic immediately:

 Â term 1 term 3 2x 5 x -1

Â Â  (x - 1)(2x + 5)

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