I am reading page 42 of a revision booklet from 2015 called GCSE AQA Mathematics Higher Level, Complete Revision Guide and Practice, by CGP..
The topic is completing the square. Below is a copy of the question and the step-by-step instructions given by the booklet. I am having trouble following, so I hope someone here can explain each step.
Fractions are written as 5/2 etc...
Write 2x² + 5x + 9 in the form a(x + m)² + n
2x² + 5x + 9
2(x² + 5/2x) + 9
2(x + 5/4)²
2(x + 5/4)² = 2x² + 5x + 25/8
2(x + 5/4)² + 47/8 = 2x² + 5x + 25/8 + 47/8
= 2x² + 5x + 9
So the completed square is: 2(x + 5/4)² + 47/8
Please follow the same method my book does, even if it is not the most elegant.
$(x+m)^2$ will have the form $x^2+2mx+m^2$. Notice that $x^2$ is alone, no coefficient.
So the first step is removing the coefficient from the $x^2$ term.
If we write $2x^2 + 5x + 9$ as $2(x^2 + 5/2x) + 9$, then we've accomplished that.
The term in the bracket looks like $x^2+2mx+m^2$ but is missing the $m^2$ part.
If $2m=5/2$; that means $m=5/4$ and $m^2=25/16$
Thus we can write it as
$ \begin{align} 2(x^2 + \frac52 x) + 9 &= 2(x^2 + \frac52 x + \frac{25}{16} - \frac{25}{16} ) + 9 \\ &= 2(x^2 + \dfrac52 x + \frac{25}{16}) - \frac{25}{8} + 9 \\ \end{align}$
Now notice that the term within the brackets is of the form $x^2+2mx+m^2$ which is essentially
$(x+m)^2$ with $m = \frac52$