Can someone please explain to me the steps of how to complete the square and then write the parabola y=x^2+6x+5 in the form (x-h)^2 = 4a(y-k).
Also from this how do I find the vertex, focus and the equation of the directrix.
In order to complete the square, you need some background knowledge about expanding and factorising.
Do you know how to use FOIL? If not, remember that $\displaystyle (a + b)(c + d) = ac + ad + bc + bd$.
In other words, the Firsts, Outers, Inners and Lasts have all been added together.
Now let's consider a special case.
If we expand $\displaystyle (1 + n)^2$ we would have
$\displaystyle (1 + n)^2 = (1 + n)(1 + n) = 1 + n + n + n^2 = 1 + 2n + n^2$.
Supposing we had only been given the first two terms... namely $\displaystyle 1 + 2n$.
Can you see that to make the third term, $\displaystyle n^2$, we would have to divide the middle term by $\displaystyle 2$ and then square it.
But you can't just add to something without changing the something. So what you add you have to subtract as well.
So $\displaystyle 1 + 2n = 1 + 2n + \left(\frac{2n}{2}\right)^2 - \left(\frac{2n}{n}\right)^2$
$\displaystyle = 1 + 2n + n^2 - n^2$
$\displaystyle = (1 + n)^2 - n^2$.
Can you see that we have "completed" a square?
Let's follow this same process using your example.
$\displaystyle y = x^2 + 6x + 5$
$\displaystyle = x^2 + 6x + \left(\frac{6}{2}\right)^2 - \left(\frac{6}{2}\right)^2 + 5$
$\displaystyle = x^2 + 6x + 3^2 - 3^2 + 5$
$\displaystyle = (x + 3)^2 - 9 + 5$
$\displaystyle = (x + 3)^2 - 4$.
So to complete the square, you need halve and then square the middle term. Once you have this number, add and subtract it from the equation, and you then have a perfect square.
In answer to your other question - once you have your equation in the form $\displaystyle y = a(x - h)^2 + k$, the turning point is given by $\displaystyle (h, k)$.
So for your question, the turning point is $\displaystyle (-3, -4)$.
Ok complete the square y=x^2+6x+5
in general take the number with x and divide it with 2 then square the result
(6/2)^2 = 9 then add and 9 to the both side of the equation
$\displaystyle y+9 = {\color{red}{x^2 + 6x +9}}+5 $
now you can write the formula in the red as a complete square
$\displaystyle y+9=(x+3)^2 +5$
$\displaystyle 4(y+1)=(x+3)^2$
c=4/4=1
the vertex is (-1,-3) right ..
directrix substitute x=0 in the equation the focus you sum c with the y vertex coordinate
I am late