 The general equation of a conic section is this, where a, b, c, d, e, and f are real numbers. If b is not equal to 0, the conic section is rotated, and these are messy and require special handling, so we'll initially focus on the b equal to 0 conics. Given an equation in this form, we can transform it into an equation in standard form by completing the squares on x and y. So for example, let's find the center and eccentricity of the conic section with this equation, and then sketch a graph. So we'll complete our squares, and so we'll rearrange this so our x terms are together and our y terms are together. And it will be convenient to factor out the coefficients of our x squared and our y squared. So we'll factor four from our x terms, and nine from our y terms. So to complete the square on x squared, we need to add four, and to pay for it, we'll subtract four. Similarly, we'll complete our square on y squared, do a little algebra. Now, because our square terms are added, this is some sort of an ellipse, but in our ellipse, the constant term needs to be one. So we'll divide both sides by 25. Again, for our standard form of an ellipse, we don't have a coefficient in front of the square term, so we'll multiply numerator and denominator by one fourth. Similarly, for the standard form of an ellipse, the y squared term doesn't have a coefficient, so multiply numerator and denominator by one ninth. And our denominator is usually the square of something, so we'll rewrite those denominators. And so we see that this equation corresponds to the ellipse x squared divided by five half squared plus y squared divided by five third squared equal to one, translated two units to the left and one unit upward. For an ellipse, remember the eccentricity is c divided by a, where a is the length of the semi-major axis, and c, the focal length, is a squared minus b squared. So we'll compute and find our eccentricity. So this is a translation of this ellipse, and so for this ellipse, we have vertices. And we also find some other points the ellipse passes through, so we can graph it, and then we translate it, two units to the left, and one unit upward. This moves the center to negative two one, and the eccentricity remains the same. Or we can try and find the focus and directories of the conic section with this equation and try to sketch the graph. Now since there's no y squared term, we can't complete the square on y, but we can still complete the square on x. So completing the square on x, it's probably easiest if we declutter the x terms and get rid of this constant, and we can complete the square by adding nine, and factorization. So this is the parabola eight y equals x squared, shifted three units to the right and down two units. Now remember that the coefficient of y is four p, and so p is equal to two. And this tells us that the focus will be located two units above the vertex, and the directories will be a horizontal line two units below the vertex. So if we take our parabola and shift it right three units and down two units, this will place our focus at three zero and the directories at y equal negative four. Or we can try another one. So let's complete our square. Again, it's convenient to factor out the coefficient of our x squared and our y squared. So factoring 25 from our x and x squared terms, we'll need to be a little bit more careful here. We want to factor a negative nine from our y squared and y terms. We'll complete the square, do a lot of algebra, and we get our equation of a hyperbola in standard form. And again, it's convenient to rewrite our denominators as the squares of something. So we see that this is the graph of x squared divided by three squared minus y squared divided by five squared shifted two units to the right and one unit downward. And we can find the asymptotes. We find two points the hyperbola passes through, and we can sketch our graph. We find our focal distance, and so our foci are located square root 34 away from the center in the horizontal direction. And when we translate, all features undergo the same translations. And so the foci will move too.