Completing the square
Recall that 
and 
. Thus for any expression of the form 
, we can add a number to make it a perfect square. To get this number, take half the coefficient of x and square it.
Written in symbols we have
So 
Where did the -9 come from? Look again at the working above. Half of 6 is 3, 3 squared equals 9, so we add 9 to form a perfect square. But, so that we don't change the value of the expression, we have to put a -9 into the expression to counter the 9 we have already added.
Here is another example.
The next example requires more care.
We need to work with 
Completing the square on a quadratic function aids interpretation, that is, you look at the algebra to see the geometry. Once you've completed the square you'll be able to write the co-ordinates and the nature of the turning point.
Here's how you'd sketch 
| => no real roots => the graph does not cross the x-axis |
x = 0 => y = 5=> | the graph crosses the y-axis at (0, 5) |
| y is never less than 1 y = 1 when x = 2 So there is a minimum turning point at (2, 1) |
Hence |  |
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GCSE Bitesize - Maths
Have you got the x-factor? Check out your factorising skills in this useful gsce bitesize activity.
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