SOLUTION: Please help sketch the graph and name the conic section with equation: 4x^2-16x+9y^2+18y=0 So far I have factored out the 4 and 9. What comes next?Thank you

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Question 66246: Please help sketch the graph and name the conic section with equation:
4x^2-16x+9y^2+18y=0 So far I have factored out the 4 and 9. What comes next?Thank you
Found 2 solutions by checkley71, rapaljer:
Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
4X^2-16X+9Y^2+18Y=0
4X(X-4)+9Y(Y+2)=0
+graph%28+300%2C+200%2C+-6%2C+5%2C+-20%2C+20%2C4x%5E2+-16x+%2B9y%5E2+%2B18y%29+ (graph 300x200 pixels, x from -6 to 5, y from -20 to 20,4x^2 -16x +9y^2 +18y).

Answer by rapaljer(4551) About Me  (Show Source):
You can put this solution on YOUR website!
I'm afraid the other solution that was posted for this problem is incorrect. Anytime you have an equation of a conic section with x%5E2 and y%5E2 with both having positive but unequal coefficients, it will be an ELLIPSE, NOT a PARABOLA.

If both coefficients are the same sign with equal coefficients, it will be a CIRCLE.

If coefficients are of opposite sign, then it will be a HYPERBOLA.

A PARABOLA results when there is an x%5E2 but no y%5E2 or a y%5E2 but no x%5E2.

This is the way it should be solved by completing the square:
4x%5E2-16x%2B9y%5E2%2B18y=0

You must first factor out the coefficients of x^2 and y^2 like this:
4%28x%5E2-4x%2B____%29%2B9%28y%5E2+%2B2y+%2B+_____%29=0
4%28x%5E2-4x%2B4%29%2B9%28y%5E2+%2B2y+%2B+1%29=0%2B16%2B9
4%28x%5E2-4x%2B4%29%2B9%28y%5E2+%2B2y+%2B+1%29=25

This is an ellipse. The standard form for an ellipse is in a form = 1, so divide both sides of the equation by 25 to set it equal to 1.
%284%28x-2%29%5E2%29%2F25+%2B%289%28y%2B+1%29%5E2%29%2F25=25%2F25
%284%28x-2%29%5E2%29%2F25+%2B%289%28y%2B+1%29%5E2%29%2F25=1
Finally invert the coefficients of 4 and 9 in order to write this:
%28%28x-2%29%5E2%29%2F%2825%2F4%29+%2B%28%28y%2B+1%29%5E2%29%2F%2825%2F9%29=1

That would be standard form for an ellipse! The center is at (2,-1), with the "radius" extending 5%2F2 units in the x direction, and 5%2F3 units in the y direction.
graph+%28300%2C300%2C-5%2C5%2C-5%2C5%2C+-1%2B%28sqrt%2825-4%28x-2%29%5E2%29%29%2F3%2C+-1-%28sqrt%2825-4%28x-2%29%5E2%29%29%2F3%29+

R^2 at SCC