SOLUTION: What type of conic is the equation? 3x^2+2x+15y^2-4y=30

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Question 1030904: What type of conic is the equation?
3x^2+2x+15y^2-4y=30
Found 2 solutions by josgarithmetic, Edwin McCravy:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
You want to adjust the equation into whichever standard form will fit variables and coefficients. You will probably need to Complete the Squares to do this.

3%28x%5E2%2B%282%2F3%29x%29%2B15%28y%5E2-%284%2F15%29y%29=30

, for which you need to understand the lesson on Completing the Square;

3%28x%2B1%2F3%29%5E2%2B15%28y-2%2F15%29%5E2=30%2B10%2F3%2B4%2F15

3%28x%2B1%2F3%29%5E2%2B15%28y-2%2F15%29%5E2=30%2B50%2F15%2B4%2F15

3%28x%2B1%2F3%29%5E2%2B15%28y-2%2F15%29%5E2=30%2B54%2F15

3%28x%2B1%2F3%29%5E2%2B15%28y-2%2F15%29%5E2=30%2B18%2F5

%28x%2B1%2F3%29%5E2%2B5%28y-2%2F15%29%5E2=10%2B6%2F5---------------Although finishing the equation into standard form is still not finished, you see that the expression with x and the expression with y have POSITIVE coefficient, and the constant righthand members is also positive; but the coefficients on the left member's expressions are unequal. The equation will be for an ELLIPSE.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor is right, but you don't need to 
do all that just to determine what type of conic
it is the equation of.  All you need do
is look at the terms in x² and y² and go by
these rules:

Rules for determining the type of conic an equation
of the form Ax²+Cy²+Dx+Ey+F=0 is the equation of:

1. If the coefficients of both the x² and y² terms when 
they are on the same side of the equation are EXACTLY THE
SAME NUMBER, the conic is a CIRCLE.

2. If the signs of both the x² and y² terms when 
they are on the same side of the equation are the 
SAME, the conic is an ELLIPSE.

3. If the signs of the x² and y² terms when they are 
on the same side of the equation are OPPOSITE, the 
conic is a HYPERBOLA.

4. If there is only an x² term but no y² term,
or only a y² term and no x² term, the conic is a
PARABOLA.

3x²+2x+15y²-4y = 30 

The x² term is +3x² and the term in y² is +15y². 
They have the same sign + and they are on the same
side of the equation, so rule 1 holds, and the conic
is an ELLIPSE.

Edwin