Will award a lot of points : )

the general form of a parabola is x^2−2x−8y−23=0 .

what is the standard form of this parabola?

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Step-by-step explanation:

Your final answer is the standard form of a parabola.  Since your equation has an x-squared term in it and not a y-squared term, your form will be

To get it into this form we will solve the quadratic for y and then set it equal to 0 so we can complete the square on it.  Solving for y then setting y equal to 0:

so

We only need to complete the square on the x-terms, so we will move the constant back over to the other side of the equals sign:

The rule for completing the square is that the leading coefficient HAS to be a 1.  Ours is 1/8, so we have to factor it out.  When we do that we are left with:

To complete the square on the left, we take half the linear term, square it, and add it onto both sides.  Our linear term is 2 (the number stuck to the x-term).  Half of 2 is 1, and 1 squared is 1.  So we add it into the parenthesis on the left.  BUT we cannot discount the 1/8 sitting out front there.  It is a multiplier.  So what we actually added on the left is 1/8(1).  That looks like this:

Now we will write the left side into its perfect square binomial (which was the whole reason for doing this!) and simplify the right at the same time:

Now we will set the whole thing back to equal y:

That's one form.  But you need it in vertex form, so we add 3 to both sides:

and then multiply both sides by 8:

If you need to break it down further to include what your p value is, then:

Either that one or the one right above it should work.