Question

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

Answer 1

`y=19/7(x+9/19)^2+717/133`

Explanation:

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

`y=a(x-h)^2+k`

The vertex can be pulled from this form as `(h,k)`.

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

`y=19/7 x^2+18/7 x+ 6`

Step 2. Factor out `19/7` to get `x^2` alone.

`y=19/7(x^2+7/19xx18/7+7/19xx6)`

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

Step 3. Simplify your terms

`y=19/7(x^2+18/19x+42/19)`

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`: `18/19`
Cut it in half: `1/2xx18/19=9/19`
Square the result: `(9/19)^2=81/361`
Finally, add and subtract that term inside the parenthesis:

`y=19/7(x^2+18/19x+color(red)(81/361)-color(red)(81/361)+42/19)`

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

`y=19/7(color(blue)(x^2+18/19x+81/361)-81/361+42/19)`

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

`y=19/7(color(blue)((x+9/19)^2)-81/361+42/19)`

Combine the remaining two fractions inside the parenthesis.

`y=19/7((x+9/19)^2+717/361)`

Step 5. Multiply the `19/7` back through to each term.

ANSWER: `y=19/7(x+9/19)^2+717/133`

So the vertex is at `h=-9/19` and `k=717/133` which can be expressed as

`(-9/19, 717/133)~~(0.4737,5.3910)`