Question

Identify the vertex, focus, and directrix of the parabola with the equation x^2 – 6x – 8y + 49 = 0

Answer 1

complete the square first

Answer 2

how would i do that?

Answer 3

You have the quadratic expression x^2 - 6x. You want to add onto it a value that results in something that factors into (x - a)^2. We could call the value to add on, c.
x^2 - 6x + c = (x - a)^2 = x^2 - 2a x + a^2 -2a = -6 comparing like-terms coefficients and the value to add on is c = a^2 from the a you find..

Then you subtract the same value so that you have in total added AND subtracted the same value, thus a net change of 0 to the equation.

Answer 4

(x^2 - 6x + c) - c - 8y + 49 = 0
^ this part in parentheses will become the expression in the form (x - a)^2.
The c outside the parentheses is just collected with the other constants.