Identifying a Conic

To decide whether a general conic is an ellipse, parabola or hyperbola, we only need to completer the square. The form of the completed square will determine which type of conic it is.
Example: Which type is the conic  
\[x^2+4x+2y^2+6y=5\]
,
Completing the square for the  
\[x\]
  coordinate gives the equation
\[(x+2)^2-(2)^2+2y^2+6y=5\]

Completing the square for the  
\[y\]
  coordinate gives the equation
\[(x+2)^2-(2)^2_2+2(y+3/2)^2-2(3/2)^2=5\]

Sim[ifying the equation gives  
\[(x+2)^2+2(y+3/2)^2=(x+2)^2+(\frac{y+3/2}{\sqrt{2}})^2=27/2\]

This is the form of an ellipse. The axes of the ellipse are  
\[x+2=0\]
  and  
\[\frac{y+3/2}{\sqrt{2}}=0\]
, and the major axis is  
\[\frac{y+3/2}{\sqrt{2}}=9\]
.

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