# What is the vertex of y=x^2-x+9-2(x-3)^2 ?

Oct 8, 2017

$\left(\frac{11}{2} , \frac{85}{4}\right)$

#### Explanation:

Simplify to $y = a {x}^{2} + b x + c$ form.

$y = {x}^{2} - x + 9 - 2 {\left(x - 3\right)}^{2}$
Use FOIL to expand $- 2 {\left(x - 3\right)}^{2}$
$y = {x}^{2} - x + 9 - 2 \left({x}^{2} - 6 x + 9\right)$
$y = {x}^{2} - x + 9 - 2 {x}^{2} + 12 x - 18$
Combine like terms
$y = - {x}^{2} + 11 x - 9$

Now that we have turned the equation to $y = a {x}^{2} + b x + c$ form,
Let's turn them to $y = a {\left(x - p\right)}^{2} + q$ form which will give the vertex as $\left(p , q\right)$.

y=-(x^2-11x+?)-9+?

To make perfect square like ${\left(x - p\right)}^{2}$, We need to find out what ? is.

We know the formula that when ${x}^{2} - a x + b$ is factorable by perfect square ${\left(x - \frac{a}{2}\right)}^{2}$, we get the relationship between $a$ and $b$.

$b = {\left(- \frac{a}{2}\right)}^{2}$

So $b$ becomes ? and $a$ becomes $- 11$.

Substitute those values and let's find ?.

?=(-11/2)^2
?=(-11)^2/(2)^2
∴?=121/4

Substitute ?=121/4 to y=-(x^2-11x+?)-9+?

$y = - \left({x}^{2} - 11 x + \frac{121}{4}\right) - 9 + \frac{121}{4}$
$y = - {\left(x - \frac{11}{2}\right)}^{2} - \frac{36}{4} + \frac{121}{4}$
$y = - {\left(x - \frac{11}{2}\right)}^{2} + \frac{85}{4}$

∴y=-(x-11/2)^2+85/4

Therefore, we have turned the equation to $y = a {\left(x - p\right)}^{2} + q$ form which will give our vertex as $\left(p , q\right)$

∴p=11/2, q=85/4

∴Vertex (11/2, 85/4)

Oct 8, 2017

$\left(5.5 , 21.25\right)$

#### Explanation:

This equation looks scary, which makes it hard to work with. So, what we're gonna do is simplify it as far as we can and then use a small part of the quadratic formula to find the $x$-value of the vertex, and then plug that into the equation to get out our $y$-value.

At the end, there's this part: $- 2 {\left(x - 3\right)}^{2}$

Which we can factor to $- 2 \left({x}^{2} - 6 x + 9\right)$ (remember it isn't just $- 2 \left({x}^{2} + 9\right)$)

When we distribute that $- 2$, we finally get out $- 2 {x}^{2} + 12 x - 18$.

Put that back into the original equation and we get:

${x}^{2} - x + 9 - 2 {x}^{2} + 12 x - 18$, which still looks a bit scary.

However, we can simplify it down to something very recognizable:

$- {x}^{2} + 11 x - 9$ comes together when we combine all the like terms.

Now comes the cool part:

A small piece of the quadratic formula called the vertex equation can tell us the x-value of the vertex. That piece is $\frac{- b}{2 a}$, where $b$ and $a$ come from the standard quadratic form $f \left(x\right) = a {x}^{2} + b x + c$.

Our $a$ and $b$ terms are $- 1$ and $11$, respectively.

We come out with $\frac{- \left(11\right)}{2 \left(- 1\right)}$, which comes down to

$\frac{- 11}{- 2}$, or $5.5$.

With knowing $5.5$ as our vertex's $x$-value, we can plug that into our equation to get the corresponding $y$-value:

$y = - {\left(5.5\right)}^{2} + 11 \left(5.5\right) - 9$

Which goes to:

$y = - 30.25 + 60.5 - 9$

Which goes to:

$y = 21.25$

Pair that with the $x$-value we just plugged in, and you get your final answer of:

$\left(5.5 , 21.25\right)$

Oct 8, 2017

Vertex $\left(\frac{11}{2} , \frac{85}{4}\right)$

#### Explanation:

Given -

$y = {x}^{2} - x + 9 - 2 {\left(x - 3\right)}^{2}$

$y = {x}^{2} - x + 9 - 2 \left({x}^{2} - 6 x + 9\right)$
$y = {x}^{2} - x + 9 - 2 {x}^{2} + 12 x - 18$
$y = - {x}^{2} + 11 x - 9$

Vertex

$x = \frac{- b}{2 a} = \frac{- 11}{2 \times \left(- 1\right)} = \frac{11}{2}$

$y = - {\left(\frac{11}{2}\right)}^{2} + 11 \left(\frac{11}{2}\right) - 9$

$y = - \frac{121}{4} + \frac{121}{2} - 9 = \frac{- 121 + 242 - 36}{4} = \frac{85}{4}$

Vertex $\left(\frac{11}{2} , \frac{85}{4}\right)$