Identify each equation without completing the square.
$9x^2+25y^2-54x-200y+256=0$

Verified step by step guidance

1. Recognize the given equation: \(9x^2 + 25y^2 - 54x - 200y + 256 = 0\). This is a quadratic equation in two variables, which suggests it might represent a conic section.
2. Identify the coefficients of \(x^2\) and \(y^2\). Here, \(9\) is the coefficient of \(x^2\) and \(25\) is the coefficient of \(y^2\). Since both coefficients are positive and different, the equation represents an ellipse.
3. To confirm it's an ellipse, check that the equation is in the general form of a conic section: \(Ax^2 + By^2 + Cx + Dy + E = 0\). Here, \(A = 9\), \(B = 25\), \(C = -54\), \(D = -200\), and \(E = 256\).
4. Since \(A\) and \(B\) are both positive and \(A \neq B\), this confirms the equation is an ellipse.
5. Note that completing the square is not necessary to identify the type of conic section, but it would be required to find the center and axes of the ellipse.