# What is the vertex form of 7y = 3x^2+2x+1?

Feb 15, 2018

Vertex form is:

$y = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} + \frac{2}{21}$

or if you prefer:

$y = \frac{3}{7} {\left(x - \left(- \frac{1}{3}\right)\right)}^{2} + \frac{2}{21}$

#### Explanation:

Given:

$7 y = 3 {x}^{2} + 2 x + 1$

Divide both sides by $7$ then complete the square:

$y = \frac{3}{7} {x}^{2} + \frac{2}{7} x + \frac{1}{7}$

$\textcolor{w h i t e}{y} = \frac{3}{7} \left({x}^{2} + \frac{2}{3} x + \frac{1}{9} + \frac{2}{9}\right)$

$\textcolor{w h i t e}{y} = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} + \frac{2}{21}$

The equation:

$y = \frac{3}{7} {\left(x + \frac{1}{3}\right)}^{2} + \frac{2}{21}$

is pretty much vertex form:

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

with multiplier $a = \frac{3}{7}$ and vertex $\left(h , k\right) = \left(- \frac{1}{3} , \frac{2}{21}\right)$

Strictly speaking, we could write:

$y = \frac{3}{7} {\left(x - \left(- \frac{1}{3}\right)\right)}^{2} + \frac{2}{21}$

just to make the $h$ value clear.