SOLUTION: Good Morning, Can anyone show me the steps for this, please? Consider the following equation and determine if it is a parabola, an ellipse, or a hyperbola and sketch it's graph:

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Good Morning, Can anyone show me the steps for this, please? Consider the following equation and determine if it is a parabola, an ellipse, or a hyperbola and sketch it's graph:       Log On


   

Question 968527: Good Morning, Can anyone show me the steps for this, please?
Consider the following equation and determine if it is a parabola, an ellipse, or a hyperbola and sketch it's graph:
4y^2+6x−2y+3x^2−15=2y^2+4x^2
Thank you in advance for your help.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
4y%5E2%2B6x-2y%2B3x%5E2-15=2y%5E2%2B4x%5E2
Does this solution process contain a mistake?
3x%5E2%2B6x%2B4y%5E2-2y-15-2y%5E2-4x%5E2=0
-x%5E2%2B6x%2B2y%5E2-2y-15=0
x%5E2-6x-2y%5E2%2B2y%2B15=0
Complete The Squares for x and y
x%5E2-6x%2B9-2%28y%5E2-y%29%2B15=9
%28x-3%29%5E2-2%28y%5E2-y%2B1%2F4%29%2B15=9%2B%28-1%2F2%29
%28x-3%29%5E2-2%28y-1%2F2%29%5E2%2B15=8%261%2F2
%28x-3%29%5E2-2%28y-1%2F2%29%5E2=8.5-15=-6.5
-%28x-3%29%5E2%2B2%28y-1%2F2%29%5E2=6.5
-%28x-3%29%5E2%2F6.5%2B%28y-1%2F2%29%5E2%2F%286.5%2F2%29=1
Possible mistake but at least steps still lead to hyperbola with vertex on the y axis but not on the x-axis.

Sign mistake, MAYBE made.
A way to avoid that, if it actually happened, could be to start with all x on one side and all y on the other side, and then perform Completion of the Squares.

4y%5E2%2B6x-2y%2B3x%5E2-15=2y%5E2%2B4x%5E2
4y%5E2-2y-15=-6x-3x%5E2%2B4x%5E2
4y%5E2-2y-15=x%5E2-6x
4%28y%5E2-%281%2F2%29y%29-15=x%5E2-6
4%28y%5E2-%281%2F2%29y%2B1%2F4%29-15-4%281%2F4%29=%28x%5E2-6x%2B9-9%29
4%28y-1%2F2%29%5E2-15-1=%28x-3%29%5E2-9
4%28y-1%2F2%29%5E2-16=%28x-3%29%5E2-9
4%28y-1%2F2%29%5E2-%28x-3%29%5E2=-9%2B16
4%28y-1%2F2%29%5E2-%28x-3%29%5E2=7
%284%2F7%29%28y-1%2F2%29%5E2-%28x-7%29%5E2%2F7=1
%28y-1%2F2%29%5E2%2F%287%2F4%29-%28x-7%29%5E2%2F7=1-----Without checking through a solution a third time, this one is probably correct for less likely to have made arithmetic mistakes.
Still hyperbola with vertices on the y axis IF were in a form centered at the origin; but this one is centered at ( 7, 1/2 ).

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
Good Morning, Can anyone show me the steps for this, please?
Consider the following equation and determine if it is a parabola, an ellipse, or a hyperbola and sketch it's graph:
4y^2+6x−2y+3x^2−15=2y^2+4x^2
Thank you in advance for your help.
After combining like-terms, it's obvious that the leading coefficients (+ 1 and - 2) have different signs. 

Therefore, it's a highlight_green%28HYPERBOLA%29.