Math Snap

STEP 1

What is this asking?
We've got this funky equation, and we need to figure out if it's a circle, hyperbola, ellipse, or parabola, and then rewrite it in its snazziest form!
Watch out!
Don't get tripped up by the big numbers!
They're just there to make it look tough.
Also, remember that the order of terms in the equation doesn't tell us what kind of conic section it is.

STEP 2

1. Rewrite the equation
2. Complete the square
3. Identify the conic section
4. Write in standard form

STEP 3

Let's group those $x$ terms together to get ready for some completing-the-square action!
We'll move everything to one side to make it easier to see what we're working with.
$$0 = -23x^2 + 644x + y - 4547$$

STEP 4

Now, let's factor out that $-23$ from the $x$ terms.
This will help us complete the square perfectly!
$$0 = -23(x^2 - \frac{644}{23}x) + y - 4547$$

STEP 5

To complete the square, we take half of the coefficient of our $x$ term, which is $\frac{1}{2} \cdot (-\frac{644}{23}) = -\frac{322}{23}$, and square it: $(-\frac{322}{23})^2 = \frac{103684}{529}$.
This is our magic number!

STEP 6

We add and subtract this magic number inside the parentheses to keep our equation balanced.
$$0 = -23(x^2 - \frac{644}{23}x + \frac{103684}{529} - \frac{103684}{529}) + y - 4547$$

STEP 7

Now, we can factor that perfect square trinomial we've created!
We'll also distribute the $-23$ to the added and subtracted magic number.
$$0 = -23(x - \frac{322}{23})^2 + \frac{23 \cdot 103684}{529} + y - 4547$$ $$0 = -23(x - \frac{322}{23})^2 + y + \frac{2384732}{529} - 4547$$$$0 = -23(x - \frac{322}{23})^2 + y + \frac{2384732 - 2400763}{529}$$$$0 = -23(x - \frac{322}{23})^2 + y - \frac{160431}{529}$$

STEP 8

Since we only have one squared term ($x^2$), we know this equation represents a parabola!

STEP 9

Let's isolate $y$ to get our equation into standard form.
$$y = 23(x - \frac{322}{23})^2 + \frac{160431}{529}$$

The equation represents a parabola, and its standard form is $$y = 23(x - \frac{322}{23})^2 + \frac{160431}{529}.$$