### Instruction: Which type of conic section is described by the equation \[x^2 = 2x + 2y - 3?\]Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above.### Response: To identify the type of conic section, I need to rewrite the equation in a standard form.
I can start by completing the square for the x terms.
I subtract 2x from both sides and get \[x^2 - 2x = 2y - 3.\]
Then I add the square of half of the coefficient of x, which is \((-2/2)^2 = 1\), to both sides and get \[x^2 - 2x + 1 = 2y - 3 + 1.\]
I factor the left side as a perfect square and simplify the right side and get \[(x - 1)^2 = 2y - 2.\]
I can divide both sides by 2 to get \[(x - 1)^2/2 = y - 1.\]
Now I can see that this is a parabola, since it has the form \[(x - h)^2 = 4p(y - k),\] where (h, k) is the vertex and p is the distance from the vertex to the focus.
The vertex of this parabola is (1, 1) and p = 1/2.
The parabola opens upward, since the coefficient of x is positive.
Therefore, the answer is P for parabola.