# What is the vertex form of y= (x-1)(x – 6) ?

Jun 26, 2017

$y = {\left(x - \frac{7}{2}\right)}^{2} - \frac{25}{4}$

#### Explanation:

Let's convert this into standard form. Then we can "complete the square" to solve for the vertex form.

$y = \left(x - 1\right) \left(x - 6\right)$

$y = {x}^{2} - 6 x - x + 6$

$y = {x}^{2} - 7 x + 6$

Now let's complete the square. To do that, we need to find a value that make ${x}^{2} - 7 x$ a perfect square. To find that value, we take the middle term, $- 7$, and we divide it by $2$. That gives us $- \frac{7}{2}$. Now we square the fraction: $\frac{49}{4}$

Now we have the value that makes the equation true. But!! we cannot introduce a new value! Not without immediately subtracting it, which would make the final value $0$.

$y = {x}^{2} - 7 x + 6 + \frac{49}{4} - \frac{49}{4}$

So, we added $\frac{49}{4}$ and then $- \frac{49}{4}$. Now let's rearrange it so we have a perfect square.... and other stuff:

$y = {x}^{2} - 7 x + \frac{49}{4} + 6 - \frac{49}{4}$

Let's rewrite ${x}^{2} - 7 x + \frac{49}{4}$ as a perfect square: ${\left(x - \frac{7}{2}\right)}^{2}$

Now our equation is $y = {\left(x - \frac{7}{2}\right)}^{2} + 6 - \frac{49}{4}$

combine like-terms

$y = {\left(x - \frac{7}{2}\right)}^{2} - \frac{25}{4}$