Can someone please show me how to complete this square?
I think there should be a way to factorize using a square identity, leaving alone a constant based on (since you are completing the square) :
I know what follows is not very mathematic but perhaps it could be a good idea to try with some numbers substituting and and investigate what happens, if a pattern doesn't show up ... but you have to find working numbers ...
Here is what I would do :
I would say , therefore :
Woah ! A square identity !
And I successfully completed the square.
Try with :
Erm ... we have to do some working there :
(substract 7 from both sides)
And then the magic operates :
Thus, it seems that regardless of what value takes, we always have , where is dependent on .
Basically, we know that (since we always take off the left-hand side just the needed amount to leave on the left-hand side)
Which simplifies to : , and finally, .
Now let's put it all together, and detail the steps more mathematically (what was shown up there was brainwork) :
But we need a little 1 on the left-hand side to successfully complete the square :
I think that is what the book wants