# How do you write the Vertex form equation of the parabola  y = x^2 - 10x +17?

Mar 16, 2016

$\textcolor{b l u e}{\text{Vertex form } \to y = {\left(x - 5\right)}^{2} - 8}$

#### Explanation:

This process introduces an error that has to be compensated for
For a really detailed explanation of the process see my solution
http://socratic.org/s/asNAQ6ru

Different values but the method is sound
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let $k$ be a corrective constant

Given:$\text{ } y = {x}^{2} - 10 x + 17$

$\text{ } y = {x}^{2} - 10 x + k + 17$

Insert the first 2 terms into brackets

$y = \left({x}^{2} - 10 x\right) + k + 17$

Take the power (index) outside the bracket

$y = {\left(x - 10 x\right)}^{2} + k + 17$

Apply $\left(\frac{1}{2}\right) \times 10 x$ and get rid of the $x$

$y = {\left(x - 5\right)}^{2} + k + 17$

'~~~~~~~~~~~~~~~~~~~ Note ~~~~~~~~~~~~~~~~~~~~~~~~~~
The introduced error is from the ${\left(- 5\right)}^{2} = + 25$ when you square the bracket. So $k = - 25$

$\text{ } {\left(x - 5\right)}^{2} = {x}^{2} - 10 x \textcolor{red}{+ {\left(- 5\right)}^{2}}$

The $\textcolor{red}{+ {\left(- 5\right)}^{2}}$ is not in the original equation!!!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$y = {\left(x - 5\right)}^{2} \textcolor{red}{- 25} + 17$

$\textcolor{b l u e}{\text{Vertex form } \to y = {\left(x - 5\right)}^{2} - 8}$

$\textcolor{m a \ge n t a}{\text{You can see from the graph that the two equation produce the same plot.}}$ 