what should be done in splitting the middle term method?

 

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:

2x²  =  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:

term 1	term 3
2x	-1
x	5

 2x , 5 + x , -1 = 10x - x
                                                       = 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.

 We have not chosen the correct factors.  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:

term 1	term 3
2x	5
x	-1

   (x - 1)(2x + 5)