# What is the vertex form of 7y = 19x^2+18x+42?

May 25, 2017

$y = \frac{19}{7} {\left(x + \frac{9}{19}\right)}^{2} + \frac{717}{133}$

#### Explanation:

Strategy: Use the technique of completing the square to put this equation in vertex form:

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

The vertex can be pulled from this form as $\left(h , k\right)$.

Step 1. Divide both sides of the equation by 7, to get $y$ alone.

$y = \frac{19}{7} {x}^{2} + \frac{18}{7} x + 6$

Step 2. Factor out $\frac{19}{7}$ to get ${x}^{2}$ alone.

$y = \frac{19}{7} \left({x}^{2} + \frac{7}{19} \times \frac{18}{7} + \frac{7}{19} \times 6\right)$

Notice we just multiply each term by the reciprocal to factor it out.

$y = \frac{19}{7} \left({x}^{2} + \frac{18}{19} x + \frac{42}{19}\right)$

Step 4. For the term in front of $x$, you must do three things. Cut it in half. Square the result. Add and subtract it at the same time.
Term next to $x$: $\frac{18}{19}$
Cut it in half: $\frac{1}{2} \times \frac{18}{19} = \frac{9}{19}$
Square the result: ${\left(\frac{9}{19}\right)}^{2} = \frac{81}{361}$
Finally, add and subtract that term inside the parenthesis:

$y = \frac{19}{7} \left({x}^{2} + \frac{18}{19} x + \textcolor{red}{\frac{81}{361}} - \textcolor{red}{\frac{81}{361}} + \frac{42}{19}\right)$

The part that can now be expressed as a perfect square is in blue.

$y = \frac{19}{7} \left(\textcolor{b l u e}{{x}^{2} + \frac{18}{19} x + \frac{81}{361}} - \frac{81}{361} + \frac{42}{19}\right)$

This gives you the perfect square using the number you got when you cut it in half (i.e., $9 / 19$)

$y = \frac{19}{7} \left(\textcolor{b l u e}{{\left(x + \frac{9}{19}\right)}^{2}} - \frac{81}{361} + \frac{42}{19}\right)$

Combine the remaining two fractions inside the parenthesis.

$y = \frac{19}{7} \left({\left(x + \frac{9}{19}\right)}^{2} + \frac{717}{361}\right)$

Step 5. Multiply the $\frac{19}{7}$ back through to each term.

ANSWER: $y = \frac{19}{7} {\left(x + \frac{9}{19}\right)}^{2} + \frac{717}{133}$

So the vertex is at $h = - \frac{9}{19}$ and $k = \frac{717}{133}$ which can be expressed as

$\left(- \frac{9}{19} , \frac{717}{133}\right) \approx \left(0.4737 , 5.3910\right)$