# What is the the vertex of y =x^2-6x-7 ?

Jun 8, 2017

$P \left(3 , - 16\right)$

#### Explanation:

There are different ways this can be done.

This equation is in standard form, so you can use the formula $P \left(h , k\right) = \left(- \frac{b}{2 a} , - \frac{d}{4 a}\right)$ Where the (d) is the discriminant. $d = {b}^{2} - 4 a c$

Or to save time, you can find the (x) coordinat for the vertex with $- \frac{b}{2 a}$ and put the result back in to find the (y) coordinat.

Alternatively, you can rearrenge the equation into vertex form:
$a {\left(x - h\right)}^{2} + k$

To do this start by putting a outside the brackets. This is easy because $a = 1$

${x}^{2} - 6 x - 7 = 1 \left({x}^{2} - 6 x\right) - 7$

Now we have to change ${x}^{2} - 6 x$ into ${\left(x - h\right)}^{2}$
To do this we can use the quadratic sentence: ${\left(q - p\right)}^{2} = {q}^{2} + {p}^{2} - 2 q p$

Let's say $q = x$ therefore we get:
${\left(x - p\right)}^{2} = {x}^{2} + {p}^{2} - 2 x p$

This looks sort of what we need, but we are still far of, as we only have ${x}^{2}$.

If we look at ${x}^{2} - 6 x$, we can se that there is only one part raised to the power of two, therefore ${p}^{2}$ must be removed. This means:

${\left(x - p\right)}^{2} - {p}^{2} = {x}^{2} - 2 x p$

Looking at the right side, we can see it is almost ${x}^{2} - 6 x$, in fact we only have to solve $- 2 x p = - 6 x$ $\iff p = 3$

This means:
${\left(x - 3\right)}^{2} - 9 = {x}^{2} - 6 x$

Another way of doing it would be to make a qualified guess and use the quadratic sentences to see if it is correct.

Now go back to our original formula and replace ${x}^{2} - 6 x$ with ${\left(x - 3\right)}^{2} - 9$

We get:

$1 \left({x}^{2} - 6 x\right) - 7 = 1 \left({\left(x - 3\right)}^{2} - 9\right) - 7 = 1 {\left(x - 3\right)}^{2} - 9 - 7 = 1 {\left(x - 3\right)}^{2} - 16$
This is similar to the vertex form:
$a {\left(x - h\right)}^{2} + k$
Where
$h = 3$ and $k = - 16$

When the quadratic equation is in vertex form, the vertex is simply the point $P \left(h , k\right)$

Therefore the vertex is $P \left(3 , - 16\right)$